Skew polynomial rings and nilpotent derivations

SKEW POLYNOMIAL RINGS

AND NILPOTENT DERIVATIONS

SKEW POLYNOMIAL RINGS

AND NILPOTENT DERIVATIONS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUW-KUNDE, VOOR EEN COMMISSIE U I T DE SENAAT TE VERDEDIGEN OP WOENSDAG 21 J U N I 1967 TE 14 UUR

DOOR

THEODORUS HERMANUS MARIA SMITS

geboren te 's-Gravenhage

D I T PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. H. J. A. DUPARC

C O N T E N T S

Introduction 9 CHAPTER I The calculus

§ 1 Multiplication 13 § 2 Basic rules 14 § 3 Linear units 19 § 4 The nilpotent a-derivation d 20

§ 5 The equation 9^2 = 991e (or 930 = 95(6/^)) 23 CHAPTER II The polynomial units

§ 6 Quadratic units 25 § 7 Units of degree (r—1) 31 § 8 The units x'^ay'^ andjv"ax" 33 § 9 The general decomposition theorem on units . . . . 35

CHAPTER III Structure of the polynomial ring R

§ 10 The inner degree 40 § 11 Prime polynomials 41 § 12 The prime factorization 43 CHAPTER IV Structure of a field K with a nilpotent a-derivation d

§ 13 The quadratic case 47 § 14 The basis theorems 49 § 15 Examples of nilpotent derivations 55

References 59 Samenvatting 61

I N T R O D U C T I O N

T h e well-known formula of analysis [ab)' = ab-\-ab' may be written in an alternative operational form

D{ab) = {Da)b+a{Db). Similarly one has

A{ab) = {^a){Eb)+a[^b) = {Aa)b + iEa){Ab),

where A denotes the difference operator and E the shift operator. Both formulas give rise to the following algebraic definition.

Let K he a (skew) field *) with endomorphisms S and T; an [S,T)-derivation of /T is a linear mapping A oï K into itself which satisfies

{ab)A = aA-bS+aT-bA {a,b £ K),

where operators are written on the right of the element on which they operate and where the operator A has no more the special meaning it has in analysis. An (5,1)-derivation is also called an S-derivation (cf. JACOBSON [20], p. 170 or CoHN [9], p. 534).

Given a field K, an cndomorphism a and a (1,(7)-derivation 6 of K, we denote by K[x; a, f5] the ring of skew polynomials in an indeterminate x subject to the commutation formula

x-a={aa)x + ab {a e K). „ Every element of this ring can be written uniquely in the form N A;<**A:^

/ = 0 {k^^i e K, polynomial of degree n) and the multiplication is completely deter-mined by the commutation formula. This ring has been first described by O R E [24], who showed that the ring X[x;cr, 5] satisfies a right division algo-rithm, hence every left ideal of this ring is principal (left principal ideal domain, left P I D for short). Because of the fact that every element (polynomial) has a finite prime factorization we conclude by [10, Theorem 5.5, Corollary 1] that this ring is a unique factorization domain (UFD for short). The notion of a U F D is called to mind by the following definitions:

1. two elements a,b of a ring R are said to be similar, if RjaR ^ RjbR, as right *) T h e term 'field' will be used in the sense of'skew field', i.e. 'not necessarily c o m m u t a t i v e division ring'.

i?-modules; in an integral domain (ring with a unit-element and without zero-divisors) this impHes RjRa ^ RjRb (cf. JACOBSON [19], p. 33 or COHN [10], p. 314);

2. a U F D is an integral domain such that every nonunit has a factorization into primes; two different factorizations of the same element have the same number of prime factors and the factors are similar in pairs (cf. COHN [10], p. 317).

In this thesis we consider the ring R of polynomials over the (skew) field K in a single indeterminate x with the commutation formula

x-a = aix-\-a2x'^^...-\-arX\ at e K (r = 2 , 3 , . . . ) ,

where the at depend on a. By the associative and distributive laws we obtain many relations between the mappings (5^: a^ at {i = l , 2 , . . . , r ) . If we assume that 02, ds,...,dr are right ^-independent, then these relations can be simplified. In Chapter I I I we derive that R satisfies a right Euclidean algorithm, hence R is a left P I D . Because of a finite prime factorization i? is a U F D (Theorem 8, §12).

From the relations obtained by the associative law it can be derived that if a is the mapping c -> Ci, i.e. ci = ca, then a is an cndomorphism of K and 02'.c^ C2is an (a^,a)-derivation of X.

Assume further a is an automorphism of K with inverse p and put «2 = ada, then d is an a-derivation of/T. In Chapter I we derive a/c = aè'^^'^a {k=\,2,...), hence from 0 = «r+i = aö''a we observe that Ó is a nilpotent a-derivation of K and the index of nilpotence of Ö is r.

It is rather remarkable that j = x^i satisfies again the linear Ore-rule ya = {aji)y — afiè (Theorem 2).

Thus we can obtain R = K^x] in the following way. Take any skew poly-nomial ring K\^y;li, —^30] with è^ = 0, adjoin j ^ i = x and take subring of polynomials in x. The rings R = /^[A:] and R = K[j; fi, —fid] have the same quotient field Q = K{y; fi, —/8(5). We can also obtain K[x] as follows. Take the skew polynomial ring L = K[t;a'',0], then R = L[x] with x'' = t {R/L is generated by x).

In Chapter II we investigate the polynomial units o( R = K^x]. First we give examples of splitting up units as products of units of lower degree. For this purpose we have to solve many ('differential') equations in the field K with the nilpotent a-derivation ó. In § 9 we derive in a few lines the general decompo-sition theorem of units. Every unit can be written as a product of units of degree

< r—\, for r ^ 2 even as a product of units of degree < r—2. The group G of units is multiplicatively generated by r isomorphic fields of units Ki of the form y/C^;* [i = 0, l , 2 , . . . , r — 1 ) . For r = 2 the ring R is the free product of

K and Ki = yKx (Chapter IV, Theorem 9). If a = 1 and r is equal to the characteristic ( > 3 ) oï K, then the ring i? is a proper homomorphic image of the free product of the fields K,Ki,K2,...,Kr-i over the constant field C (Theorem 6).

In § 11 we obtain relations between prime polynomials in K\^x] and in K\j; /3, —^(3]. It is still an open problem to derive the fact that A'[A:] is a P I D only from the fact that K[y; /9, —fid] = K[x-'^; /S, -,9(3] is a P I D .

In Chapter IV we deal with the structure of a (skew) field K with a nil-potent a-derivation d of index r. For r = 2 the field K turns out to be a non-commutative quadratic extension of the constant field which has been de-scribed by COHN [9]. For arbitrary r we derive two basis theorems which appear to be similar to theorems on cyclic fields derived by ALBERT [1], PERLIS [25] and AMITSUR [3]. In § 15 we investigate examples of a (skew) field K of charac-teristic p^O and a derivation ( = 1-derivation) nilpotent of index r = p*^. Examples of nilpotent «-derivations (a ^ 1) were already given in § 4.

CHAPTER I

T H E C A L C U L U S

§ 1 Multiplication

Let K denote an arbitrary commutative or non-commutative field with an ar-bitrary characteristic. The objects of our investigations are polynomials in a formal variable x

F{x) = a(o)x» + a<i)A:»-i + . . . + «(»), (1) with fl*"' belonging to K. Let

G{x) = è(o);c™ + é(i)Ar'™-i + . . . + *(»") (2) be a second polynomial of the same kind. Sum and difference F{x) ±^ G[x) are

defined as usual i.e. by the polynomial one obtains from (1) and (2) by adding or subtracting corresponding coefficients. The polynomial cF{x), where c is an arbitrary element oï K, is the polynomial obtained from F{x) by multiplying all coefficients on the left with c. The polynomials (1) therefore form an additive Abelian group with K as domain of multipliers.

We shall now define multiplication for the additive group formed by the polynomials (1), so that the group becomes a ring. We assume that the multi-plication of polynomials is associative and both-sided distributive. It is clear that, due to the distributive property, it suffices to define the product of two monomials bx^-ax^, or even more specifically, to define the product x-a. We assume the commutation formula

x-a = arX^-\-ar-\X'^~^-^...-^a2X^^aix^ao ( r = l , 2 , . . . ) , (3) where «r,«r-i,...,flo are elements of K depending on a. From (3) one easily obtains

x{a + b) = xa + xb = {ar~\-br)x^+{ar-i + br-i)x''-'^ + ... + {ai-{-bi)x+{aü + ba). This leads to the following relations

{a + b)i = at + bi, z = 0,1,2,...,r, (4) hence the mapping èi:a-^ ai (z = 0, l , . . . , r ) is an cndomorphism of the

additive group of the field K.

I n the special case of a skew polynomial ring ^ [ X ; ( T , (5] we have (3j = 0 [i > 2), (5i = CT and (3o = h.

T h e special properties of the mappings hi will be discussed further in the following section.

§ 2 B a s i c r u l e s

T h e principal formula (3) yields

x^a = {xar)x''-\-{xar-i)x''^'^-\-...-\-{xao) =

= aoo+{aio+aoi)x+{a2o + an + ao2)x^ + ...-\-arrX^^ = = 0(2,0)+0(2,1 )^ + «a2)-ï^ + ---+'Z(2,2r)^^',

where

^(2,/) a^j. Similarly a,^,, = 2 ^ (z,.^,._ ;^,

We obtain by induction on k: ann = } an kr x'^a = 2 ^ a(t,i)X«. (5) 1=0 Because of <^(t,i) = 0 for 2 > ^'^ (6) formula (5) can be rewritten in the form

CO

x^a = ^ a,t_,.,A;«. (7)

1=0

Important relations are found from the associative law {xa)b = x[ab), all a,h e K. One has by (7)

CO CO

(A:a)é = N a.x'è = ƒ a^b^^j^x^, 1=0 i,i = 0 and x((ïè) = N {ab)iX^ = y {ab)ixK 1=0 1=0 Equate coefficients: r ((Zé), = ^ (z,é(,,, (f = 0 , 1 , 2 , . . . ) , (8) /t = 0

where for simplicity b(o,i) = (3oi-è, with dot the Kronecker delta. For i > r the left-hand side of (8) vanishes, hence

r

t = 2

To be able to continue the calculations we successively require three assump-tions (C), (A) and (B).

ASSUMPTION ( C ) . Iföi'.a^- ai then (52,...,(5r are right K-independent, i.e.

(a(52)<r(2> + («(53)<:(3) + . . . + («ór)<:('-> = 0 (allueK), (10) implies c<2) = c(3) = . . . = cf) = 0.

This assumption is satisfied for instance if for each / there exists an element y<'' e K such that

vmi ^ 0, v(i)öm = 0 (m > /), / = 2,3,...,r.

In fact if (10) holds, let e**' be the first non-zero coefficient and put u = z^(*>, then

0 = (ö(«>(5i)c<«) + 0 (zero because z;<»>ó; = 0 forj' > i). Now we have ii<*)(3j 7^ 0, <;<*> ^ 0, hence a contradiction.

Because of assumption (C) relation (9) yields immediately

è(,,., = 0 for^ = 2 , . . . , r ; z = r + l , r + 2 , . . . (11) Of course from

"(k,r+\) = "{k,r + 2) = • • • = "(jt./tr) ^ ^

it follows

" ( t + l . S r + l ) = ^ ( / : + l , 2 r + 2) = " • • • ^ "{k+\,kr+r) = 0 ,

so that the relations (11) are a consequence of the matrix relation

B*^{b,,,,] = Q {k = 2,...,r;i = r+\,...,2r). (12) Now (5) becomes xi'a = 1 = 0 .T Y^a^,,^xi, k=\,2,...,r. (13)

With vector ^ = {\,x,x'^,...,x^) and yl the matrix («(i,,;) ik,i = 0,1,...,r), we can rewrite (13) in the form ^-(2 = Ax. From x{ab) = [xa)b = A{xb) = ^ 5 x , we see that if A -> /I and b ^ B, then ai -^ ^ 5 . We observe that a -> («(t,,)) = ^ is a matrix representation o[ K.

Further we can derive

CO oo CO CO

xi'ix'a) ^x^Y^ a^,-,xi = J^ {x''a^,Jxi = J^^j '^('.•H*.i.)^*^^'

1=0 1=0 1 = 0 ; - 0 CO CO x'ix^a) = ^ ^ fl(i,y, (/,,-,^*+^, 1=0 j = 0 CO xMa = 2 a(i+,,„„x™, m=0

so that equating the coefficients of x™ we conclude

i+j~ m i+j — ni

Now we introduce a very radical simplification.

ASSUMPTION (A), «o = Ofor all elements a e K {so èa = 0).

T h e assumption (A) leads further to (Zj^,, = 0 for i < k, so (13) becomes

xka = Y^a^,;,x\ k=\,2,...,r. (15)

i = k

From (15) we conclude that the polynomial ring R has no zero-divisors. T h e relations (8) now also become simpler, namely

{ab)i. = 2_^a),b^k,if, i=\,2,...,r. (16) * = i

E.g.

{ah) I = aibi, (17) {ab)2 = «2^11 + 01^2, (18) {ab)3 = a3bin + a2{b2i + bi2)+aibs, (19)

{ab)r = arb(r,r)+ar-lbir-l,r) + ... + aibr. (20) Clearly a:a ^ «i is an cndomorphism of/C and a(r,r) = ^a''. Then (15) says

x''a — {aa^)x^.

If a = (3i would be the zero cndomorphism, then it easily follows from (16) that Ó2 = (^3 = . . . = (5r = 0. This contradicts assumption (C), so a 7^ 0, i.e. a is necessarily a monomorphism of A".

A further assumption (B) will appear important. ASSUMPTION ( B ) . a is an automorphism of K, with inverse p.

Often we write «(-D for afi, a(-2) for aft"^ etc. We shall also introduce the map-ping (3 by

ah = fl2(-i), hence 02 = aha. (21) In virtue of (18) h satisfies the condition

{ab)d^ {ah){ba)+a{bh), a,h e K, (22) in other words, h is an a-derivation of K. T h e constant field C is defined as the

subfield of (5-constants, so

Now we continue with the basic rules. With (14) we derive / , '^(r-l,i)(l,,-) = ^ <^(l,i)(r-l,i) = ^(r,r+k) = 0 ('ï = 1 ; 2 , . . . ) , i+j = k+r i+i = k + T SO by (11) this yields ^(r-l,r) { l , t ) + ' ^ ( r - l , r ~ l ) (1,*+1) = "^(1,*) (r-l,r)+^(1,A+1) ( r - 1 , r-1) ^ 0 , or '^(i—l,r)A = ^ ' ^ ( r - l . r - l ) (* + l)) ( 2 3 ) and «/t(^-l,r) = - % + l ) ( r - l , r - l ) ( ^ = 1 , 2 , . . . ) (Aj = . . .) . ( 2 4 ) By (11) it is obvious that ^ ~ "[r,r+\) = "(r-\,r)\~\~b(,-\,r-\)2 = " l ( r - l , r ) + " 2 { r - 1 , r—1)> hence

é(r_-i,r) = —ba^^'^h a n d *i(r-i,r) = —bha^. We derived

é(^_,^^j = -bar-^ = -bpda'-. (25) Combining (24) and (25) we obtain

— ajcfiha^ = — a*:+ia'-i, so

a*+i = akfiha = a\{phaY = a\fih''a = ah^a. Conclusion:

flft+i = a(3*^a (A = 0 , 1 , 2 , . . . ) . (26) We shall often use (26) in the form

am(-l)n = am+n-l, (27) where m and n are positive integers.

In virtue of Or+i = ad^a = 0, we see (5'" = 0, i.e. ó is a nilpotent a-derivation of index r.

With (27) it is not difficult to understand the identity

^(k,i) = '^(i,;-i) (-i)2+'^(t-i,z-i)i • (28) With the help of this formula we can easily derive that if the first column of the

matrix B* of (12) is zero, then the other columns follow automatically. So the fundamental basic relations are (27) and

*(2,r+l) = *(3,r+l) = . . . = ^.,r+l, = 0. (29) They are in fact consequences of {ab)r+i = {ab)6^a = 0, i.e. (29) is a

conse-quence of the nilpotency of h. From (14) we can obtain further

thus

^(r,r) (l,i) = <^l\,k}{r,T) Of flf,,,)* =

ü^^r.r)-By (26) we can rewrite the relation

(l(r,r)k = «Mr,r) (30)

in the form

aar^t-i = a^i-iar^ k = 1,2,..., consequently

a'-ó = har. (31) Relation (31) follows also from (25). T h e results obtained may now be

summed up as follows:

T H E O R E M 1. Let R be the ring of'skew' polynomials in x, consisting of all polynomials 2a<*>;c* {where a<*' belongs to the {skew) field K), with the usual addition and with a multiplication which is defined by

{A) x-a = aix-\-a2X^-'r...'j-arX'', r = 2, 3, ..., then

{i) the mapping a: a ^> a\ is an cndomorphism of K. Under the further condition

{B) a is an automorphism of K, with inverse /5, one has:

{ii) the mapping h defined by «2 = aha is an a-derivation of K, satisfying (22). If the ring R satisfies the conditions {A) and {B) and moreover the condition

(C) the mappings hi.: a-^ at {i = 2,3,...,r) are right K-independent, then one has

{Hi) hn = (3"^ia, an = ah"~^a {higher a-derivations), n = 1,2,3,...;

{iv) dr = 0, (3''~i 7^ 0, i.e. h is a nilpotent a-derivation of index r. This implies e.g. a^h = hW.

{v) The relations induced by the associative law can be reduced to a(i.r+i) = 0, i = 2 , 3 , . . . , r ; they follow also directly from the nilpotency of h.

Often we use the derived relations

((z-'). = aT\ (32) ((z-')2 = -(zr'(2,a„', ((z-')(5 = - a - ' ( a ( 5 ) ( a « ) ^ ' , (33)

((z~')3 = ar'(«2^ri'«(2,3)—^3)«ril> (34)

( ^ )(2,3) = C'W Ö{2,3)'^111) V'^^) ( ' ï * ) (11-1.1.) = « C n - l , n ) * ( . , i , ) + « ( i , - l , n - l ) * ( n - l , » ) ' ( 3 6 )

hence the mapping a ^ <^(ii-i,ii) i^ ^'^ (a",a"~i)-derivation of A^. Finally we have from (36)

~ ' ^ ( n - l , i i - l ) ' ^ ( i ! - l , i i ) ' ^ ( n , i i ) - \^') I " ^(11-1,11)

§ 3 Linear u n i t s We can write

{x—ai){x-\-a) = arX''-\-ar~ix''-''--\-...-\-a3X^-ir{a2+'\.)x^—aia = = —aia in the case (Z2 = — 1 .

We observe that (x-\-a) with 02 = —1 is a linear unit. Now it is interesting to consider the equation

«2 = — 1 or ah = —I. From (24) we obtain with k = I

aur^i,r) = —a2a''-^, so «2 = — 1 iff ai(r-l.r) = 1, (38) a(~i)2 = — 1 iff «(r-l,r) = 1. (39) Now assume ^ = {^{2,r)'^i- }(-!) = ^(2,r) {-l)<^r(-l) with c an arbitrary element o£ K{cr 7^ 0 or ch''"'^ 7^ 0).

We have by (18) and (33): -1 „ , - 1 ,

{^(2,r+l) ^(l,r)lKl '•(2,r)''r ^r+l''r = - ^ . I ^ i l ' = - 1 ,

where the second equality holds in virtue of (28) and (27). Apparently by (38):

To verify this directly we first need some other relations. If we put nz = 2r in (14), assume A:,/ < r and remind (29), then we have immediately the for-mula

«(i,r) (w = %+;,2r) {k,l<r). (40) By (36) the verification is now straightforward:

{''{2,rfr }(l—l,r) = <^(2,r) (r-l,r)<^r(r,r)+^(2,r) ( r - 1 , r-1) (^r )(r-I,r) = ( U S e ( 3 7 ) )

•^C^f^ r\ CfOrWr— I r—n^r(r—1 r — H ^rl'r — 1 rl^r/

In virtue of (?,(,,., = C(i,r) (r,r) = ^{r+i,2r) etc. by (40) this becomes: ^(r+l,2r)''(r+l,2r) ^(2,r) ( r - I , r - l ) ^ r ( r - 1 , r-l)''(r,2r)^r(r,r) = ^^ Conclusion: One has

{'^(2,r) (-l)'^r(-l)}2 = ~ 1 (^I) and

for all c G A with Cr 7^ 0. It is also easy to verify that

{(M'-i)-i(M'-i-*)}(3* = 1 {k = \,2,...,r~l), (43) for allb eK with hhr-^^ ^ 0.

§ 4 T h e nilpotent a-derivation 6

We begin with the introduction of the symbol jv = x^^ and the rather remark-able result:

THEOREM 2. Let R = K[x] be the polynomial ring defined by the assumptions {A), {B) and [C) of Theorem 1, then the indeterminate y = x~^ satisfies the commutation formula

ya = {aP)y-afih {for all a e K). (44) Proof. Combining (15) and (25) we obtain

x'^-ifl = ((za'-i)x''-i —(fla''-ió)A:'' (45) and

x>-b = {ba'')x'', (46) hence ya = x'^a = x''-'^{x-ra) = xr-'^{{afi'')x^r} =

= {A;'-I((Z/?'')}X-'" = {(a/3)x'-i —(a/?(3)x'-}x-'- =

= {afi)y-{aph).

COROLLARY 2.1. In order to get the general case of (3) with (Zo ^ 0 and subjected to the conditions of Theorem 1, take any skew polynomial ring K[y; P, —Ph] with h^ = 0, adjoin y^^ = x and take subring of polynomials in x.

Conversely given this, we have ya = afiy — afih = aft{y — h), thus with J = x~i

ax = xa/3(l—(5x), hence xa = ax{l—hx)~'^a =

= ax(l+(3x+(32x2 + . . . + ,5'-ix'--i)a =

= ((za)x+ ((Z^a)x2-f ((Z(32a)x^ + . . . + {ahr-^a)xr.

Further we have symbolically (using again the rules hx = xh, ax = xa) xay = ai + (Z2^ + ...+ö!r*''~"^ =

= (aa) + (a(5a)x + . . . + (a^'-ia)x'"-i = = 41+(5x+((3x)2 + . . . + (^x)'--i}a =

hence from xaby = xay-xby it follows that

((zé)(l-(3x)-ia = a ( l - ( 3 x ) - i a - è ( l - ( 3 x ) - i a . (48) This expresses the fact that (1 —hx)~^a is an isomorphism of the field K; in fact

it is the automorphism generated by the derivation h ( = infinitesimal auto-morphism) by the Taylor formula.

If we want to have examples of the mappings a^ at (higher a-derivations) we have to look for nilpotent a-derivations of the field K. First we have the following result:

THEOREM 3. Let K be a {skew) field with a derivation h which is nilpotent of index r, then K has characteristic p ^ 0 and r ^ pK

Proof. Leibnitz' formula learns

0 = {ab)hr = y j M(aó'-*)(é(3*) for all a,b e K.

k=i ' '

In virtue of (43) we can choose elements b such that

è(3'^ 7£: 0, è(3*+i = 0, /t = l , 2 , . . . , r - l . It follows that

n((Z(3'-'^-) = 0 for all a e K, k= l , 2 , . . . , r ~ l ,

hence

From L ^ r = 0 we conclude that K has prime characteristic, p say, which must satisfy jölr. Write r = qp*,p-\-q. It is easy to prove that

(;•) ^ c;)

is congruent to q mod p, hence q = 1 and r = pK Shorter proof ( C O H N ) : Again following Leibnitz

«\ 0 = {ab)h'- = r{ah''-'^){bh)+iV\{ah'-'2){bh2 choose b such that bh ^ 0, bh^ = 0, then

r{ad'--i){bh) = 0 for all a e K,

we get a derivation D = hv = h^ii with index q, so p\q. Consequently q = I, r = p'' and the theorem follows again.

It is not difficult to find examples of a nilpotent derivation. E.g. let K be the field of Laurent series with finite principal part, integer coefficients modulo p {p prime) and ordinary differentiation h. Then of course h'P = 0. More gene-rally for the case r is a prime power we shall give in § 15 examples of fields of characteristic p ^ 0 with a derivation which is nilpotent of that prime power index. However we shall give already three examples of nilpotent a-derivations with a 7^ 1.

1. The simplest example we obtain perhaps if we take for K the field of com-plex numbers and define

ah = Im a, aa = a (conjugate oï a),

or with the notations of {21): a2 = Im a, a\ = a. T h e solution (43) of the equation a2 = ah ^ \ would become here

a = {ch)~^c = {Im c)~'^c = {Im c)~^{Re c)^i

for a\\ c e K with Im c ^ 0. Indeed we see clearly that ah = Im a = 1. Remark. If we choose (Zo = —a^ ^ — Im a we obtain

(X —z)2 = x^^XZ —z'x—1 =

= X2 — ( x 2 — Z X — 1 ) — z ' x — 1 = 0 .

That's why we assumed «o = 0, namely to avoid zero-divisors.

2. Let a be a generator of the cyclic group of automorphisms of a cyclic field of characteristic p ^ 0. Let a^ = 1. If we write h = a—l it is readily verified that h is an a-derivation. Because of aP = {d-\-l)J' = 1 we obtain h^ = 0. The field of constants is the set of elements left fixed by a. Cf. AMITSUR [3], p. 88.

3. Let K = F{t) he the field of fractions of the commutative polynomial ring F[t] over a commutative field F which contains a primitive zzth root of 1 (say w). We define (cf. COHN [13], p. 420)

ta = (ot, to = (1—uii)t^, and l a = f, | ó = 0 for all | e i^. We observe that ha = wah and easily prove by induction

t^hi = (1 - co«) (1 - w«+i)... (1 - a>«+^-i) ti+i (49) forz = 0, ± 1 , ± 2 , . . . a n d j = 1 , 2 , 3 , . . .

By (49) we have t^h^ = 0, hence ó is a nilpotent a-derivation of index zz. In virtue oï t^h = 0 we have for the constant field C: C = F{t^).

§ 5 T h e e q u a t i o n <p2 = (pie ( o r cpS = q»(E/S))

L e t K he a (skew) field w i t h a n a - d e r i v a t i o n h satisfying (22), w h e r e a is a n a u t o m o r p h i s m of K w i t h inverse fi. W e a b b r e v i a t e a g a i n ai = aa, at =

aha,..., aic = ah^-^a. W e ask for a n e l e m e n t (p e K, such t h a t

q>2 = <pie, (50)

w i t h e a given e l e m e n t oï K. I n view of (17), (18) a n d (27) w e successively d e r i v e from (50)

(fa = <P2{-1)2 = {<PE(~1))2 = 9'2ei + 9?lÊ(-l)2, so

953 = 9 ' l ( £ e i - f £(-1)2)

a n d

<P3{-i) = 9'(e(~i)fi + e(-i)2(-i)).

I n t h e s a m e w a y w e h a v e

<Pi = 9'3(-i)2 = (9?(e(-i)e + e ( - i ) 2 ( - i ) ) } 2 =

= 9!'2(£ieii + e(-i)2i)+9'i{(e(-i)e)2 + e(-i)2(-i)2} = = 9'2(eieii + e(-i)2i)+9'i{e(-i)2eii + e«2H-e(-i)3} = = 9?i (eeifiii+££(-1)21 + e ( - i ) 2 e i i + £«2 + £(-1)3), so

9'4(-i) = 9'(£(-i)e£i + e(-i)e(-i)2 + £(-i)2(-i)«i + e(-i)e2(-n+£(-i)3(-i)). W e a b b r e v i a t e (Pn(-l) = (pS''-^ = (pfn-l{e), (51) h e n c e / i ( £ ) = £(-1), (52) /2(£) = £(-l)£ + £(-l)2(-l), (53) /3(£) = e(-l)e£l + £(^)£(-l)2 + £(-l)2(-l)£l + £(-l)£2(-l)+£(-l)3(-l). (54) I t is easy to derive t h e r e c u r s i o n f o r m u l a f o r / n ( £ ) : 9'n+2 = 93(«+l)(-])2 = Wfn{£)}2 = = <P2{fn{s)}n+<pi{fn{e)}2 = 9 ' i [ « { / » ( e ) } " + ( ƒ » ( « ) l a ] ' c o n s e q u e n t l y ƒ.+!(£) = {/«(e)}2(-l)+£(-!){ƒ«(£) } l , (55) w h i c h m a y be r e w r i t t e n as ƒ„+!(£) = [-{fn{e)Ufn{e)-^}2 + e]i-i){fn{e)}i. (56) N o w a s s u m e / « + i ( £ ) = 0, ƒ«(£) 7^ 0, t h e n it follows from (56) t h a t {Me)-'}2 = {/„(e)-i}i£, a n d w e o b s e r v e t h a t (50) h a s a solution ip = {/«(fi)}"^.

Assume further that q> and q> are two solutions of 9^2 = 9'ie, and write

(p{0)~^ = 0, then we obtain immediately

02 = 9'2<^ri'—'Pi^r'wü' = Vi^Vn^ —9i£?u = 0,

so that Ö is a constant and 99 = 0^ = 6{fn{e)]-^ is a general solution of (50). Further is <P(n+l)(-l) = (pfn{e) = 0{fn{e)}-]fn{e) = 0, or {ö/«(e)-i}(«+i)(-i) = Ö, 02 = 0. If we choose Ö = 1 we obtain {fn{s)-'}in^l, = 1.

Altogether we have proved

THEOREM 4. Let K be a {skew) field with an a-derivation h satisfying (22) and a an

automorphism of K with inverse fi. If we write ak = ah^^^a {k = 1,2,...) and consider the equation 992 = (fie with e a given element of K, then we can derive

ipM-i) = 9fn-i{e) {or q)d«-'^ = (pfn~\{e)), where the fi{s) are given by the recursion formula

\fn+i{e) = {ƒ«(«)}2(-i)+£(-!){ƒ«(£)}i, n = 0 , 1 , 2 , . . . ,

\Me)-=L

(ffn+i{e) = 0 andf„{e) 7^ 0, then <p2 = (pie has the general solution

V = ö{/»(e)}-S Ö 6 C {i.e. O2 = 0), (57) and

{fn{e)y'h-=l ( r z = 1,2,...). (58)

COROLLARY 4.1. If h is a nilpotent a-derivation of index r andfr{e) 7^ 0, then the

equation q>2 = 9?i£ has only the solution cp = Q. Iffr{e) =fr-i{s) = . . . =/i+i(£) = 0, fi{e) 7^ 0 (z = 1,2,..., r— 1), then we have the solution

V = {fim-'- (59)

The special properties of the recursion formula (55) and the functions/«(E) will not be discussed further, although they are very interesting, especially for a = 1.

We conclude with the useful remark, that if 952 = 9'i£ and f = (p~^, then tp satisfies the equation

CHAPTER II

T H E P O L Y N O M I A L U N I T S

§ 6 Quadratic u n i t s

Let R = K[x] be the polynomial ring satisfying all the conditions of Theorem 1. In this chapter we investigate the polynomial units of R. T o find quadratic units we write

{x+y){ax^ + bx+c) = s {y,a,b,c,s G K), (61) so

{xa)x^-{-{xb)x-\-{xc)-\-yax'^-\-ybx-\-yc = s, hence by the commutation formula (A) of Theorem 1 we obtain

(arX'-+2 + (2r-lA:'-+l+ . . . ) +(6rX'-+l + èr-l^'-+...)+(CrX'^ + C r - l X ' - l + . . . ) + -\-yax'^-\-ybx-\-yc = s.

Equating coefficients of powers of x we obtain

I

bi+C2+ya = 0, (62)

ci^yb = 0.

From the equation «1 + 62 + ^3 = 0 we obtain successively

(Z + M + CÓ2 = 0,

(Z(3«-ia + é(5«a+(;ö«+ia = 0, z = 2, 3, ..., ai + bi+i+ci+2 = 0,

so (62) yields the independent equations

«1 + ^2+^3 = 0, (63) ^ bi+C2+ya = 0, (64)

Ci+yb = 0. (65) Put c = f-1, then equation (65) gives b = — (fiy)"!, and from (64) we obtain

a= -y-^{bi+C2) = - 7 - H - ( f i i y i ) - ^ + (^-^)2}. Using (33) this becomes

« = ( y . r ) " ' ( i + n f r ' f 2 ) f n ' , (66) and we have

ax' + bx+c = ( r . } ' ) " ' ( l + 7 . f r ' f 2 ) ^ n ' ^ ' - ( f i y ) " ' ^ + r ' , so

Now we have still to satisfy equation (63). T h e equation (63) 'nearly' follows from (64), because (64) gives

b+C2(-i) + {ya)(~i) = 0, hence by (27)

b2 + C3+{ya)(-i)2 = 0,

and after comparing with equation (63) we conclude that (63) may be replaced by the equation

{ya) (-1)2 = «1, so with {ya)(-i) = ip we obtain for f

Wi = y f V i i . (68) (cf. (60) where we met the same equation).

In view of (66) we have the solution

f = y(-i)a(-i) = y - i ( l + y f - i | 2 ( - i ) ) l r ' , thus

W = ( l i y ) - i + | - i | ^ 2 ( - i ) f r ' (69) and subsituting this %p in (68) we find further restrictions on the elements y and

f. We obtain after long calculations the relation

l 3 - ( l 2 + l ) f n ' ( f 2 . + ^ . 2 + l) = { ( f r " ' ) , - l K n ' ( ^ i 2 + f2.) +

+ (fr)~'(y2+i)7iV-(l2+i)fiV. (70) \ï y, ^ G K satisfy this identity, the expression (67) will be a quadratic unit.

We have

( ^ + y ) ( y . 7 ) " ' { ( y . f r ' ^ 2 + i ) f n ' ^ ' - y . ^ r ' ^ + r i 7 r ' } = y r \ (71) where y and | satisfy (70).

If we choose | = y we obtain the simpler case

( ^ + y ) ( y , y ) ^ ' { ( y 2 + i ) y u ' ^ ' - ^ + y . } = 1, (72) with y satisfying

y3 = (y2+i)yn'(y2i+yi2+i). (73) It is interesting to derive (72) in an other way. Therefore we require a lemma.

LEMMA 1. Let y e K satisfy y.^ = (y2+l)yri'(y2i+712+1); then

y» = (y2+i)yiVy(2,„) {n = i,5,...). (74) Proof We have successively

y3 = ( y 2 + i ) y n ' ( y 2 i + y i 2 + i ) > y3{-i) = (y2(^i) + i)yr'(y2+yi2(-i) + i ) ,

ïi = y3(-i)2 = {(y2(-i) + i)yr'(y2+yi2{-i) + i)}2 =

= {(y2(-i) + i)yr'}2(y2ii+yi2i + i) + (y2+i)yri'(y22+yi3) = = {y3ynl + (y2+i)(yr')2}(y2ii+yi2i + i) + (y2+i)yri'(y22+yi3) =

= {y3yul-(y2+i)yri'yi2yru}(y2ii+yi2i + i) +(72+i)yü'(722+713) =

= {73-(y2+i)yn'yi2}yril(72ii+7i2i + i) + (72+i)7ii'(y22+yi3). If we substitute 73 = ( y 2 + l ) y i i ' ( 7 2 i + y i 2 + 1 ) ; we obtain

74 = (72+i)7ri'(72i + i)7rn(72ii+yi2i + i) + (y2+i)yri'(y22+yi3) = = (y2+i)yri'y3i + (y2+i)yri'(y22+yi3),

where we used (73) again. So finally we have y* = (72+l)7ri'(73i+722+7i3) = (72+l)7n'7(2,4). Now assume (74) to be true for certain n, then we have

7i.+ l = 7r,(-l)2 = {(y2{-l)+l)7r'7(2,i,)(-l)}2 =

= {(y2{~i) + i)yr'}2y(2,«)i + (y2+i)yri'7(2,„)(-i)2 =

= {ysynl —(72+l)7n'yi27ill}7(2,.)i + (72+l)7n'y(2,„)(-i)2 = = {y3—(72+l)y^I'7l2}y^lly{2,„)l + (72+l)7u'7{2,„){-l)2• Substituting again 73 = (72 + l)7n'(72i+7i2 + l) we have

y.+i = (y2+i)yri'(y2i + i)yrily(2,n)i + (y2+i)yri'y{2,ii)(-i)2 = = (y2+i)yri'7«i + (y2+i)yri'y(2,»)(-i)2 =

= (72+i)yri'{7(i,ii)i+7(2,11) (-1)2}-By (28) we have finally

7i,H = (72+l)7ri'7(2,n + i)> and the lemma follows by induction on n.

From this lemma it is not difficult to obtain in an other way the general quadratic unit with a linear complementary unit x + y . We write

{dx^+ex+f){x+y) = t {d, e,f y,teK), so

d{x^y)-\-e{xy)-\-dx^+ex^+fx = t—fy.

After equating the coefficients of powers of x we obtain the following equations for d, e and f.

dy(2.i)+eyi = 0, z = 4, 5 , . . . , r; (75)

</(y2i+yi2+l)+«y3 = 0; (76) dyn + e{y2+l) = 0; (77) eyi+f=0. (78)

From the equations (76) and (77) we conclude

73 = -e~'d{y2i+yri+l) = ( 7 2 + l ) 7 r i ' ( 7 2 i + 7 i 2 + l ) . By Lemma 1 we have also

y.- = (y2+l)7ri'7(2,i), i = 4, 5 , . . . , r, so that the equations (75) are dependent on (76) and (77).

It remains to solve

idyn+e{y2+l) = 0 , \eyi+f=0.

Hence

dx^^ex-\-f = —e(72+l)7n'x^ + «x~e7| = = -«{(y2+i)yri'-*^'^--«+7!}, and just as in (72), (73) we conclude again

{ ( 7 2 + l ) 7 i A ' ^ - ^ + 7i}(^ + 7) = 7i7> where y satisfies

(72+Oyfl'(712+721 + 1) = 73. Now we derived in two ways the quadratic unit

(72 + l ) 7 n ' ^ ' " ^ + 7i (79) (y satisfying (73)), we shall try to decompose it in a linear unit and a constant

term. Therefore we write

( ^ r ' ^ - 7 i ^ " ' ) ^ = ^ r ' ^ ^ - y i =

= g\'grX'+gV'gr-\X''^+---+gi^g2x' + x-y^, and we have to find and element ,§ e j ^ for which

gr = gr-\ = . . . = & = 0, ^r'^2 = - ( 7 2 + i ) 7 r i ' . Because ^3 = gh^a = 0 implies ^4 = gh^a = 0 etc., it remains to solve

Us = 0,

U2yii+^iy2 = — ^ 1 . By (18) this may be written

I g3 = 0, (80) I (^7)2 = - ^ 1 . (81) In virtue of the following lemma equation (80) can even be omitted.

L E M M A 2. Let g and 7 be elements of K, such that {gy)2 = —,^i and 73 = (72+l)yri'(y2i+712+1), then g^ = 0.

Proof. {gy)2 = —^1 imphes (^7)3 = —,^2, hence by (19) , ? 3 y i U + ^ 2 ( y 2 1 + 7 l 2 + l ) + ^ l 7 3 = 0,

thus

^ r W i i i + ^ r ' & ( y 2 i + y i 2 + i ) + y 3 = O- (82)

Because (^7)2 = —^1 implies g{'^g2 = — ( 7 2 + l ) 7 n ' we can deduce from (82)

^r'^37iii~(y2+i)yil'(y2i+7i2+i)+73 = o, so

^f'^37,11 = 0, ^3 = 0 and the lemma follows.

We can still remark that with this method it is not difficult to give an alter-native proof of Lemma 1. Let

(^7)2 = - ^ 1 , then by (27) we may write

{gr)n = -gn~l {n> 3), by (16) this may be written

n

giya.n) = —gn-i {n> 3); 1 = 1

by Lemma 2 however we know ^3 = ^4 = . . . = ,^» = 0, hence ,?27(2,n)+^17» = 0, or ^r'^27(2.„)+y. = 0, hence -(y2+l)7ri'7(2,li)+7n = 0> thus 7i, = (72+l)7n'7(2,ii)> and Lemma 1 follows again.

After all it remains to solve equation (81)

(^7)2 = - ^ 1 . (83) If we put gy = (p equation (83) can be written in the alternative form

9^2 = -9^17,"'. (84) To obtain a solution for g we can solve equation (83) as well as equation (84).

First method. Consider (^7)2 = —,^1 or ^2 = ^ i { ^ ( y 2 + l ) 7 r i ' } with ,^3 = 0. Applying Theorem 4 we have the solution

g = -{L{E)V = -«("-!) with £ = -(72+1)7,1', thus

^ = 7i(l+72(-i))^i. (85) Second method. Consider 9^2 = —9'i7r') so £ = —y^^.

It turns out thaty3(£) = 0 ; in fact we have {/3(fi)}i = «eieii + ee(-i)2i + £(-i)2eii + e£2+£(-i)3 =

= - y r ' 7 ü ' 7 n l + y r ' ( y " ' ) 2 i + (y"')2ynl+yr'(yr')2-(7'"')3 = = -7r'yu'ynl—7r'yri'72i7ril-yr'y2yri'yril +

—yr'yri'yi2yril-yr'{y2yri'(y2i+yi2)-y3}7ril = = -yr'{(y2+i)7ri'(72i+7i2+i)-73}7ril = o in view of (73). Applying again Theorem 4 we obtain

f = {/2(e)}'' = {«{-i)e + fi(-i)2(-i)}^' = { 7 " ' 7 r ' - ( 7 ~ ' ) 2 ( - i ) } " ' = = { 7 ^ ' 7 r ' + 7 " ' 7 2 ( - i ) 7 r ' r ' by (33),

so

<?> = yi(l+y2(-i))"iy, (86) and

g = 997-1 = yj(l+y2(-l))-S which is the same as (85).

Now we have obtained the solution g in two ways we can write down the decomposition of the quadratic unit

( y 2 + i ) y n ' - ï ^ - ^ + y i = - ( ^ f ' ^ - y i ^ " ' ) ^ =

= { - ( y 2 + i ) y r A + y i ( i + y 2 ( - i ) ) 7 r ' } 7 i ( i + 7 2 ( - i , ) " ' . (87) Of course one might be inclined to believe that all quadratic units can be written as the product of a linear unit and a constant term. However quadratic units with quadratic complementary units cannot be decomposed in general into linear units and constant terms.

Finally we obtain from (84) the following result.

LEMMA 3. y = —<P2{-i)(p = ^('?"3)~V ^^ ^ solution of the equation (72+l)7ri'(72i+712+1) = 7 3 ,

for each cp e Kfor which 992 7^ 0 and 994 = 0 {(ph 7^ 0 and q)h'^ ~ 0).

Proof. If we write g = — 9?2(-i) we have 7 = g~^(p, so gy = cp. It follows immediately that

(57)2+^1 = 0, and

(^7)3+^2 = 0. Expanding {gy)2 and (,^7)3 we obtain

^2yii+,?i(y2 + l) = 0,

and because of ^3 = — 954 = 0 we derive from the last two equations: 73 = ( y 2 + l ) 7 i i ' ( 7 2 i + 7 i 2 + l ) , which completes the proof of the lemma. Of course we could have checked the solution for 7 directly, but because of the enormous amount of work we shall not do this here.

It is interesting to transform (87) with the solution given bij L e m m a 3. We obtain

-(pi'(P3X^—X — (p2'(Pi = —{92'x + <P2'<P\92{-i))'P2(-l}, hence

(p3X^ + <p.^x + (pi = {x+q)iq)2(^li^)(p,{-i)- (88) Of course we could have found this decomposition immediately,

O u r results may be summed up as follows:

THEOREM 5. The quadratic unit (72+l)7n'-'^^—.^^+71 satisfying

{ ( 7 2 + 1 ) 7 H ' X ^ - X + 7,}(X + 7 ) =y,y, (89) where y e K satisfies the y-equation

( 7 2 + l ) 7 n ' ( 7 2 i + 7 i 2 + l ) = 7 3 , (90) has the decomposition

{ - ( y 2 + i ) y r i ' ^ + y i ( i + 7 2 ( - i , ) 7 r ' } x 7 i ( i + y 2 ( - . , ) " ' - (91) The y-equation has a solution y = — {(ph)~^(pfor all cp & K with (ph ^Q andcph'^ = 0.

§ 7 U n i t s of d e g r e e (r—1) First we observe the following.

LEMMA 4. Every unit of degree r can be written as the product of a unit of degree {r~\) and a constant term.

Proof. If we write

{a('-)x'- + a<'-i)x'-i + . . . + a ( i ) x + (z(o)}{è(™)x™ + . . . + é(i)x + é<o)} = 1, a**), è(») e K, we have successively

{a('-)x'-é<™>+a('-i>x'-ié<™) + ....+ü:<o)i('»)} {xm + . . . + (è<'»))-iè(o)} = 1, {a('-)x'-+a<'-i)x'-i + ...+a<i)x + (j<o)}{x'» + . . . + (è('»))-ié(o)} = 1,

with new coefficients a<«) e K. Equating coefficients of x''+'" we obtain <?<''' = 0, hence

a<'-)x'- + (2<'--i)x'-i + ...+fl!(i)x+a(o) = {a<'--i)x'-i + . . . + a(i)x+a(o)}x{*<"'>}-i, and the proof is complete.

(92) Now we shall derive some units of degree (z"—1) and check whether they can be decomposed into a linear unit and a constant term. Write

(^('-i)x'--i+A('-2)x'-2 + ...+^(i)x+A<o))(x + 7) = d{deK). Equate coefficients. This yields the system of equations

yz<^-l)(l+7(r-l.r))+^<'-2'7(r-2.r) +A('-3)7,r-3,r) + • . . + + ^<2>7(2,r)+A<l)7r = 0 ^"•^l'7(r-l,r-l) +A<'-^2)(l_^y(^_2,r-l))+^<'-^3)y(r-3.r-l) + . . . + + A<2>7(2.r-l)+Aa)7r-l = 0 h^''^^y(r-2.r-2) +A('-3)(1 +7(r-3,r-2)) + . . . + + A<2)7(2,r-2)+A(l>yr-2 = 0 (93) A(2)7ii+Aa)(l+72) = 0 1^(1)71+^(0) = 0 After elimination of the elements /((*) we find that the element y must satisfy a very complicated equation. E.g. for r = 4 formula (92) becomes

[{(y2+l)7ri'(7i2+72i + l)-73}7ril^^ +

- ( 7 2 + l ) 7 n ' ^ ' + ^ - 7 . ] ( ^ + 7) = - 7 i 7 , (94) with 7 satisfying the relation

{(72+ l)7n'(712 + 721 + 1) -73}ynl (7211 +7i2i +7112+ 1) +

- ( 7 2 + i ) 7 ü ' ( y i 3 + y 2 2 + y 3 i ) + y 4 = o. (95) Returning to the general case, suppose we can find an element g e K such that

gr = 0, gr-1 = /Z(-1), gr-2 = h(r-2)^ . .., g2 = h^^\ gi = A<1), (96) then we can write the unit of (92) in the form

A(r-l);,;r-l_^^(r-2);rr-2^..._|_^(2);(.2_^^(l);,._^^(0) =

= gr-lX''^'^+gr-2X'-^ + ...+g2X^+giX + h<-0) = Xg + h<0) = xg—giyi, where we used the last equations of (93) and (96). Hence if we can find such an element^, this special unit of degree {r^l) can be rewritten as

{x—giyig^^)g = linear unit X a constant term. (97) If we combine (93) and (96) and remember (16), we obtain the following

system of equations for g

gr = 0, igy)r = -gr-h {gy)r-l = —gr-2, ,ggs (^7)3 = (^7)2 = -g2,

-gl-T h e last equation of (98) yields the other ones, so there remains to solve the equation (^7)2 = -gu (99) or -11 (100) ƒ& = ^ i { - ( 7 2 + l)7ii },

Ur = 0.

Applying Theorem 4 we have the solution

^ = [ / - 2 { - ( 7 2 + i ) y r i ' } ] - ' - (101) Alternatively we can substitute ,^7 = 99 in (99) which gives

92 = - - F i y f ' , (102) and Corollary 4.1 gives the solution

9' = U - . ( - y r ' ) r ' , (103)

^ = U - i ( - y r ' ) r V - ' - (104)

only in a factor ( — 1), but conclude by calculating the solution (104) in the case r = 4. By (54) we obtain

5 = { / 3 ( - 7 r ' ) r ' 7 - ' =

= { - 7 ' V r ' 7 n ' + 7 " ' ( 7 " ' ) 2 + (7"')2(-i)7n'+7^'(7r')2(-i)-(7"')3(-i,}~'7~' =

= -7ii{(72+i)yn'(y2i+y.2+i)-y3}r-i„ (105) which resembles very much the coefficient of x^ in (94). In fact we can prove

that if we write the unit of degree (z*—1) as

.(4x''~i + 5x''^2_|____|_^_|_2 (coefficient of x is one), then

g =-A^\ =-{AfiyK

§ 8 T h e u n i t s x"ay" a n d y"ax" Because oïy = x"i we have

(x»aj») (x"a-ij") = 1 and [y"ax"){y«a-^x«) = 1, and it is interesting to consider the units

U„{a) = x^ay« {n = 0,1,2,...) and

En{a) =y«ax'' {n = 0,1,2,...). T o begin with En we have for example (cf. (44))

Ei{a) =yax = afi — a^hx, so

E2{a) =y^ax^ = a{ph)^x^-ap{ph + hp)x + afiK (107) Further we have

Er-i{a) = y''^'^ax''-^ = xafiy =

= (Z^'-(3'-iax'-i + a/S'-ó'-2ax'-2 + ..._^a;S''(5ax + a/9'-i = = xaP''h + apr-^ =

= {x + a;S'-i(a/3'-(5)-i}zz/5'-(5, (108)

Er{a) =yraxr = afi\ (109) Er+k (a) = y^'^ax'^+k = yi^afi^x^ = Ek {afi^). (110)

In the same way we find

Ui{a) = xay = arX''~'^-'t-ar-ix''~^-\-...-\-a2X-\-ai =

= xa2(-i)+fli = xah-\-aa = {x-\-aa{ah)~'^}ah, ( H I ) U2{a) = x^ay^ = ai2.r)xr~^ + a(2,r-i)X''-^ + ...+a(2.3)X + a(2,2), (112)

Ur-i{a) = x'-iajf'-i =y{aa'')x = Ei{aa'-), (113)

Ur{a) = x'^ay'= aa"", (114) Ur+k{a) = x^+^ay+ic = x^aa^y" = Uk{aar), (115)

£„(a) =jv«zzx» = x*^'"-«(_>'*'''ax*'')7*^''-", so

En{a) = Ukr-r>{afi''r), (116) for all positive integers k such that kr—zz > 0.

Also it is easy to verify the relations

Ei{a-^a)-ah= -a-^h-Ui{a) (117) or

Ei{b)-b--^f)h = -bfih-Ui{b-^fi). (118) It is important to notice that all units of the form y^ax^ (where zz is a fixed

integer) constitute a (skew) field Kn isomorphic with the coefficient field K, this follows immediately from (48). Of course Kr = K, so that the polynomial ring R contains (z"—1) isomorphic copies Ki, K2, ..., Kr-i oïK.

Finally we shall give an example of a quadratic unit which cannot be written in general as the product of a linear unit and a constant term. Consider

Ur-2{a) = X''-^ayr-^ = a(r-2,r)X^ + a(r-2.r-l)X + air-2.r-2),

and try to write this unit as the product of a linear unit and a constant term. We have

{bx+c)d = bdrX'- + bdr-ix'-^ + ...+bd2X^ + bdix+cd, so we have to solve the equations

di = 0, i = 3, 4 , . . . , r; W2 = <Z(r-2,r);

bdi = (Z(r-2,r-l); cd = a(r-2,r-2).

It follows that our system of equations becomes j z/2 = d\{a(r-2.r-V)\~^^(r-2,r), \ dz = 0.

First we must check whether there are solutions for d unequal to the zero so-lution. From Theorem 4 it follows that ƒ2 (fi) 7^ 0 yields the unique solution z/ = 0. If/2(£) = 0, then we have the solution

d={fl{E)Y^={EP)-\ Because of

£ = {zZ(r-2,r-l)}~'^^(>--2.r) we obtain after long calculations

{/2(^)}l = fi^l+S(-l)2 = '^(r-2,i--l){'^{r-3,r-l)l^l ^(r-3,r)ll,

and we observe that £ = a(7l3,,_i,(Z(r_3,,) is the condition that there is a solution (/ 7^ 0, namely the solution

d = (fi^)-' = {a(7l2,r)«(r-2,r-l)}(-l)-Conclusion: Ur-2{a) = x'-^ay-^ = = a(r-2,r)X^ + a(r-2,r-l)X-\-a{r-2,r-2) = = [^{r-2,r)''^ + '^(r-2,r-2)^(r-2,r-l) (-l)^(r-2,r) (-l)J X Xa(r-2,r) (-l)'^(r-2,r—1) (-1),

for those elements a G K satisfying

'^(r-3,r-l)'^(r-3,r) =

^{r-2,r-l)^(r-2,r)-Hence we observe that only for special elements a G K we can decompose the quadratic unit Ur-2{a) into a linear unit and a constant term.

§ 9 T h e general d e c o m p o s i t i o n t h e o r e m o n u n i t s Assume

[a(w)x» + a(«-l)x«-l + .. .+a(l)x + (Z(0)] [},(m)xm -^h(m-l)xm-lJ^ ^, ,_f_ + bWx+b(o)] = 1,

in which zz*') does not denote a derivative, but plainly the l-th coefficient. Hence [a(0)jn + a(l)j,;»-l + . _ + a(»-l)j)) + a(n)]x»»[e(0)j);m + i(l)j))»>-l + . . .

Let A; be a positive integer such that n ^kr <c n^r, then the previous relation can be rewritten as

[a(o)j»» + a(i)jra-i + . . . + a(«-i)j + ü;<»*)]y'»'-«[é(o)a*^™ +

+ è(l>a*^JV™"'^ + . . . + è(™-l)a*^J); + è('»)a*''] =ym+kr_^ (119) where we have {m-\-kr) prime factors jv on the right-hand side. Now it follows

from a theorem of Oystein Ore ([24], Theorem 1, p. 494, or cf. JACOBSON [19], Theorem 5, p. 34) that the left-hand side of (119) can also be decomposed into the product of (z?z + A:r) linear prime factors similar to j . (For the concept of similarity see the references just quoted). Thus it follows that the polynomial

zz(o)j);« + a(i)j»*-i + ...+ö;(»-i)j + a('»)

can be represented as a product of zz linear prime factors similar to y. It is easy to prove that every polynomial similar t o j can be written as

eyf {e,feK), cf [19], p. 36. Consequently

zz(o^v» + zz(i)j»-i + . . . + a ( » - i ) j + ö;(") = p<o)j)'g(i)j);e(2)j...7p(»-i)je(«), (120) where the p(*) are certain elements oï K. After multiplication on the right with X" relation (120) can be rewritten in the form

fl("')x'» + (z(''-i)x»-i + ...+zz(i)x + a<o) =

= (?«» (J£l(l)x) (j)'2e(2)x2) . . . (j'«-lgl(»-l)x«-l) (J«p(«)x'») =

= EO{Q<°^)XEI{Q(^)) x£2<e<^') X ...x£„-i(t>(«-i)) x£'„(e(«)). (121) By (106) and (109) we observe that in the case r = 2 all the Ei{Q^^^) are ele-ments of K or linear units. For r > 3 we found in the preceding section that Eo, El, ..., Er-2 were units of degree 0, 1, 2, ..., r—2 and Er-i was a unit of degree r— 1 that could be written as the product of a linear unit and an ele-ment 0Ï K (cf. (108)). The results of § 8 and § 9 may be stated as

THEOREM 6. Let R be the polynomial ring satisfying all the assumptions of Theorem 1. Let Kn c R be the {skew) field of all units y^ax"^ {a e K;n = 0,1,2,...,r—\; Ko = K). Then Kn is an isomorphic copy of K and further

(z) every unit of R can be written as a product of units from the fields Kn',

{ii) for r > 2 every unit of R can he written as a product of units of degree < r—-2, for r = 2 as a product of linear units;

(z'z'z) the fields K, Ki, K2, ..., Kr-i generate the whole ring R;

{iv) ifr = 2, then the ring R is a homomorphic image of the free product of the fields Kand Ki;

{v) z/" a = 1 {hence K has characteristic p ^ 0, cf. Theorem 3) and r ^ p '> 3, then the ring R is a proper homomorphic image of the free product of the fields Ki {i = 0,1,...,r—1) over the constant field C.

Proof. In the beginning of this section we proved (i) and (ii). Statement (iii) follows from the fact that there are polynomial units of every degree, so that an arbitrary polynomial of R can be written as a sum of units, hence as a sum of products of elements from the Ki (z = 0, l , . . . , r — 1 ) . Statement (iv) is an immediate consequence of (iii). However statement (v) is meaningless without saying which construction of the free product we have in mind. Because we refer to the construction of C O H N [5, 6], we have to care that the family of fields Ki {i = 0,\,...,r—\) has a single amalgamated subfield. We shall prove that this is the case if a = 1 and r equals the characteristic. If a = 1, then

x-a = ax+((zó)x2 + (aó2)x3 + . . . + (aó'--i);j;r (^ g JC),

-ah {a e K). (a(3*)x*^ (z?z = l , 2 , . . . , r ) , and We and easily verify by i xmqym similarly y-a = ay-nduction

-It:

Let c e C, then c — jy'cx' = ykx^, hence

C^KiHKj {i,j = 0,1,...,r-\). We shall also prove the converse. Let e e Ki C\ Kj, e.g.

e =yibx^ =yiaxi {a,b e K; i,j = 0 , l , . . . , r — 1 ; z ^j). Suppose z' > j and put i—j = m, then

xmaym = è (1 < ZZZ < r—1), hence

a-\~m{ah)x~i-^m{m-\'l){ah^)x^->r... = h, I < zzz < p—\, so a = b and ah = 0. This yields e = a = h e C, whence

Ki n Kj ^ C {i,j = 0,l,...,r-l;i ^j).

Finally we conclude Ki 0 Kj = C {i ^j) and the family of fields Ki (z = 0, \,...,r—\) has a single amalgamated subfield, which is the constant field C. Consequently the free product of the r isomorphic copies Ki of the field K exists and the ring /? is a homomorphic image. For r ^2 {r > 2) this image is proper. To prove this, \et g G K satisfy gh = \ (such an element g exists by

(43)). We have

Writing y^'gx™' = ^('«) e Km, we obtain

^+^<i'+^<'> + . . . + ^ < ' - ' ' = rg-lr{r-\)x. Now K has characteristic p and r = p, so

^ + ^ a ) + ^ ( 2 ) + . . . + ^ ( r - l ) = 0,

which is a non-trivial relation in the polynomial ring R that does not follow from relations in the family of fields Ki. Thus R is not the free product itself but a proper homomorphic image. However for r = 2 we obtain g-\-g'-^'> =

= —X 7^ 0. Indeed in § 13 we shall verify that for z- = 2 (a = 1 or a 7^ 1) the ring R is the free product itself. This remark completes the proof of Theorem 6.

COROLLARY 6.,1. In the case r = 2 or 3 every unit can be written as a product of linear units. In the case r = 4 every unit can be written as a product of linear and qua-dratic units etc.

We shall illustrate Theorem 6 with an example. Let us try to decompose the unit {x-\-y) with

yz = ( y 2 + i ) y n ' ( y 2 i + y i 2 + i ) , (cf. Theorem 5, formulae (89) and (90)). Write

(x + 7)ö = X xpy, hence 0rX'- + 0r-lA:'-l + ...+0lX + 7e = ^>rX'^'^ + -<pr-lX''-^ + ...^-W2X^-Wl, so 61 = rpi+1, i = l , 2 , . . . , r ; yd =

t^i-If we substitute 9 = ^2(-i) in the last equation, we obtain the equation V'2 = yV^Wii- With ^p = q)~^ this becomes

<P2 = —9'iyr', with the solution (86):

9 = yi(i+y2(-i))"V, so ip = y " ' ( l + 7 2 ( - i ) ) 7 r ' , and finally (x+7)7^iy)i ^ xrpy becomes (^ + 7 ) ( 7 i 7 ) " ' ( l + 7 2 ) 7 r i ' = ^ 7 " ' ( l + 7 2 { i ) ) 7 r ' j

more comprehensible if we consider the complementary quadratic unit of ( x + 7 ) . With the notations of § 6 we can write by (87)

( 7 2 + i ) y r i ' ^ ^ - - « + y i = {-gr^x+yig-')g = = g^^{-xg+{gy)i} =

= gr\-gX-gr-iX'~^-...-giX+{gy)i}. If we use (83) we obtain

(72+l)yn'Ar^-^ + 7i = ^ r ' { ( ^ 7 ) X " ' + (^7)i-i^'"' + . . . + (^7)2^+(^7)i} = = gï'^xgyy =

= g\:^xqy.

So it is not so strange that the element <p also appeared by the complementary linear unit ( x + 7 ) .

Summary

x + 7 = X99-IJ (9917), (122) ( y i 7 ) " ' { ( y 2 + i ) y r i ' ^ ^ - ^ + y i } = ('Piy)"'^w, (123)

where 99 is given by (86). Indeed we see immediately that (122) and (123) are complementary units.

CHAPTER III

S T R U C T U R E O F T H E P O L Y N O M I A L R I N G R

§ 10 T h e inner d e g r e e Let

f{x) = a(o)x» + zz(i)x»-i + ....+zz<«-i)x + a(«), zz<») e K.

If a(o) ^ 0 the number zz is said to be the polynomial degree or outer degree of f{x). In short zz is the degree oïf{x), denoted by degf{x). If we multiply ƒ (x)

on the right w i t h j " we have

f{y) =f{x)yn = (z(«)j»+zz("-i>jv""i + ...+0(1)^ + 0(0).

Let ƒ (jc) contain n factors similar to y (for the concept of similarity see again O R E [24], p . 488 or JACOBSON [19], p. 33), then we define the inner degree zz* ofƒ(x), notation idegf{x), by zz* = zz —«. We prove

LEMMA 5. The inner degree of units is zero and a polynomial of inner degree zero is a unit.

Proof. In § 9 we derived that lïf{x) is a unit of outer degree zz, we can write (cf. (120))

f{y) = g(0)j)j^(l)j);e(2)j. ..j);g(«-l)jp(«),

with zz factors JC, so that zz* = zz — zz = 0. Conversely, ifƒ(x) is of inner degree zero, t h e n y ( j ) = / ( x ) j " contains n = n factors j just as (120), soƒ(x) is the unit given by (121).

LEMMA 6. If F{x) xG(x) = H{x), then ideg F{x) -\-ideg G{x) = ideg H{x).

Proof. Let F{x), G[x), H{x) have outer degrees /, zzz, zz and inner degrees /*, zzz*, zz*. If we multiply the relation

F{x) X G{x) = H{x)

on the right-hand side byj*''+'" where k is sufficiently large, we obtain {F{x) y^}y'"'-i[x'<:'-{G{x)y™}ykr] = {//(x)j)''»}j'^''+™-«, so

F{y)y'"--l[{G{y)}a'"-] = H{y)ylcr+m-n^

Now by counting the number of factors j on either side of the last relation one finds

{l—l*) + {kr-l) + {m-m*) = {n—n*) + {kr+m-n), hence /*+zzz* = zz*, which proves the lemma.

LEMMA 7. idegy"f{x)x'^ = ideg f {x).

Proof Put ,^(x) = y"^f{x)x™-; then x"*^(x) = ƒ(x)x™. In virtue of Lemma 6 we have

m-^idegg{x) = idegf{x)+m,

and the lemma follows. We note thaty"'f{x)x™ is a polynomial in x.

Remark. Since the sum of two units is in general not a unit, and also the sum of two non-units can be a unit, we observe that the inner degree of the sum of two polynomials has no obvious relation to the inner degrees of the two polynomials.

§ 11 P r i m e p o l y n o m i a l s

If we have a decomposition of a polynomial from ^ [ x ] into two non-unit polynomials, we know from Lemma 6 that the components have inner degrees which are less than that of the original polynomial, hence every polynomial in A^[x] possesses at least one decomposition into prime factors. There is a correspondence between prime polynomials in R = K[x] and prime poly-nomials in ^ = K[y; fi, —fih]. For this purpose we introduce the notion of j - p r i m e or semi-prime.

Definition. A polynomial in ^ = K[y; ft, —fih] is said to he y-prime or 5«zzzz'-prime if it has only one 5«zzzz'-prime factor ^y.

Let subscripts denote 'ordinary' polynomial degrees in K[x] and K[y; /?, -fid]. With these agreements we have the following lemmas.

LEMMA 8. If Hn{x) 7^ x is prime, then ffn{y) = H„{x)y is semi-prime. Proof. Assume iJ„{y) not semi-prime, e.g.

ffn{y) = My)B,n{y), l+m = zz, (124) where Ai{y) and Bm{y) both contain at least one 'real' prime factor {^y).

We can write (124) in the form

Hn{x)y = {Ai{y)xi]{yi{Bm{y)x"'}xi]yi+™,

SO

Assume that s is the least degree of any non-zero terms oï Ai{y) and that t is the least degree of any non-zero terms oï Bm{y)- Write

Ii{y)xi = Ai^s{x) and Bm{y)x"' = Bm-t{x). With these notations (125) becomes

Hn{x) = AUx){yBm-t{x)xi}. (126) We assume that Ai{y) contains a real prime factor of polynomial degree Q.

Let V be the number of factors j in ^j_s(x)y~* = ^i{y)y^. Then v < 1—S—Q,

hence

ideg Ai-s{x) = {l—s)—v > g > 1,

and we conclude that Ai^s{x) is not a unit (cf. Lemma 5), hence by (126) Hn{x) is not prime and the lemma follows.

Also the converse is true:

LEMMA 9. If Ën{y) is semi-prime, then Hn'{x) = fln{y)x^ is prime. Proof. Consider an arbitrary decomposition oï Hn'{x), e.g.

Hn'{x)=Fi{x)Gm{x). (127) We are sure that Fi{x) and Gm{x) have non-zero constants, otherwise IIn'{x)

would have right factors x and Hn{y) = II„'{x)y" would not be of degree zz, which is a contradiction. Now (127) can be rewritten as

Hn{y)x"- = {Fz(x)j'}j*'--'[x*'-{G™(x)j'»}y]x*'-+"», (128) where ^ is a positive integer such that kr—l > 0. With obvious notations (128)

becomes

iin{y) xy'"'+"' = Fi{y) xy-'x{Gm(j)a*^'-}xjv".

Now Hn{y) is semi-prime and therefore contains only one prime factor ^y, which has its similar partner either in Fi{y) or in Gm{y)a'"'. In the last case e.g., Fi{y) contains only factors similar tojv, just as in (120). Hence Fi{x) is a unit of the form (121). So in (127) either i^;(x) or Gm(x) is a unit and the lemma follows.

If ^re(^) is prime with inner degree zz*, then A„{y) = A„{x)y^ possesses (zz—zz*) factors similar t o j . For n — n* = 3 we have for example

In{y) = An{x)y» = e(0)je(l)>-e(2)^„.(j)e(3)j^(4),

where g(«) e/T and (by Lemma 8) P„*{j>) a prime polynomial in R = K[y, p, ~fih]. Consequently

so every prime polynomial An{x) of (outer) degree zz and inner degree zz* can be written (apart from left and right unit multipliers) in the formjy''P„,(x)x'. We sometimes call P„,{x) the associated minimal prime polynomial oï A„{x), i.e. a prime polynomial for which the inner degree equals the outer degree. This means that the corresponding polynomial in .^ = K[y; fi, —fid] is not only semi-prime, but also prime, so it has no factors y. Consequently every prime polynomial in R = K[x] contains a minimal prime polynomial, which corresponds with a really prime polynomial in ^ = K[y; fi, —fih]. Altogether we have proved

THEOREM 7. Let R = K[x] be the polynomial ring satisfying all the conditions of Theorem 1, then

(z) every polynomial has a decomposition into prime polynomials; {ii) if Hn{x) 7^ X is prime, then Hn{y) = IIn{x)y"- is semi-prime; (z'z'z) if fJn{y) is semi-prime, then H„'{x) = fln{y)x^ is prime;

{iv) every prime polynomial in R of inner degree n* is associated to a prime polynomial of the form y''P^t{x)x" where P„*{x) is of {outer) degree n* and also of inner degree n*.

A prime factorization in R = K [x] can be transformed into a prime factoriza-tion in ^ = K[y; fi, —fih]. In consequence of this and the theory of O R E [24] it is easy to prove that two different decompositions of the same polynomial have the same number of prime factors. Also it is possible to establish a certain equivalence between the factors in the first and in the second prime factoriza-tion. However we shall omit these proofs, because in the following section we shall derive these results in a more transparent way.

§ 12 T h e p r i m e factorization

The quadratic commutation formula (r = 2) has been met first by P. M. COHN ([9], p. 548) in a free product of two quadratic extensions. He proved that every left ideal was principal. It is not difficult to generalize his proof for an arbitrary r. In the proof we do pot need the fact that the monomorphism a is also an automorphism of K.

LEMMA 10. Let the polynomial ring R be defined by the assumptions {A) and (C) of Theorem 1, then R is a left principal ideal domain {left PID for short).

Proof. We nole first that results (iii) up to (v) of Theorem 1 cannot be proved any longer. In fact we only need formula (15). Now let / 7^ 0 be a left ideal OÏR, and let

be a monic polynomial of least degree in I. Of course ƒ is unique. Now we have Rf £. I and ' xf = arX«+'--l+((Zr_i + èr)x'»+''-2 + . . . XV = a(2.r)X«+'-l + ((Z(2,r-l)+é(2.r))A;»+'-2 + . . . X3/ = (Z(3,r)^«+'-l+(zZ(3,r-l)+é(3,r))A;» + ^-2 + . . . (129) X ' - l / = (zZ(r-l.r) + l ) ^ « + ' - l + (a(r-l,r-l)+-^(r-l,r))-ï"+''-^ + . . .

and all these polynomials belong to /. We denote by Rk the set of all polynom-ials in R of degree < k. With obvious abbreviations (129) can be rewritten as

( xf = 7iix'»+''-i + 7i2x«+'"-2 + ...+7(i''-^i>x»+i,

X2/ = y21xn + r-l_^y22xn+r-2_^___J^y(2.r-l)xn+l^ (130) (mod Rn)

. X » ' - ! / = 7('--l.l);cM+r-l_|_y(r-1.2);C»+r-2_|____j_,y(r-l.r-l);(;n+l_

First we observe that xf, xf, ..., x^-'^f ave left /T-independent modulo Rn, for if o(l)x/"+5(2);^;2^_|^..._^p(r-l);j;r-l/-= Q (mod Rn), (?(«> G K,

then since

(e(l)x + 5(2)x2 + . . . + e ( ' - l ) x ' - - l ) ) / 6 /?ƒ C ƒ, we must have

(e(i)x+e(2)x2+....+(5('-i)x'-i))/= gc)/, (131) where gC) G K. But R has no zero-divisors, hence g(i) = ^(2) = . . . = g('') = 0.

The linear system of equations (130) in the xi has one and only one solution if its determinant does not vanish (cf. O R E [23], p . 477). If the determinant of the system would vanish, then there would exist a linear relation between the right-hand sides of (130), so also between the left-hand sides of (130), which is impossible because they were left A^-independent. Hence the system (130) has one and only one solution (mod Rn) for x"+''~*', ^ = 1, 2, ..., r — 1 . E.g.

xn+r-k = Skf—qn, Sk 6 Rr-i, qn E Rn, hence

Skf= x«+'--*- + 9„. We can even obtain a relation of the form

Skf= X^+'-'^ + qn-i, Sk G Rr-1, qn-l £ Rn-1, A: = 1 , 2 , . . . , r - 1 . (132) So finally it follows that /?/"contains a monic polynomial of degree A^for every N '> n and an arbitrary element h oï R can be written as

h = qf+t, qGR, tG Rn-i. (133) I n particular iïh G I, then t G I,so t = 0 and h = qf G /?ƒ which yields / £. Rf.

From (133) it is obvious how to develop a right Euclidean algorithm in R. If we want to divide hi G R oï degree / by ^m e ^ of degree zzz, we look in the ideal Rgm for the monic polynomial of least degree, say ƒ«, so Rgm = Rfn with zz < zzz. Clearly we have ƒ„ = ugm, gm = vf„ where zz and v are units, so fn and gm are left associated. By (133) we can write hi in the form

hi = qfn + t, t G Rn-1, thus

hi = {qu)gm + t, t G Rn-1 c Rm-i. (134) We observe that deg t < n ^ m = deg gm. Notice that it is possible that I < m.

LEMMA 11. Let the polynomial ring R he defined by the assumptions {A) and (C) of Theorem 1, then R is a unique factorization domain {UFD for short).

Proof. First we show that the descending chain condition holds for left ideals having an intersection 7^ 0. Let

R ::> Ra ^ Rh => Re ^ ... ^ Rz (135) be a strictly descending chain of principal left ideals, which all contain a

fixed element 2 7^ 0. If we rewrite the ideals by means of the corresponding minimal polynomials we obtain the sequence

R ^ Ramin ^ Rbmin ^ RCmin ^ ... ^ RZmln, hence

deg ZZmin < deg hmln < deg Cmln < ... < deg Zmin

and we see that the chain (135) is finite. The ascending chain condition for left ideals follows as usual for left principal domains, thus both chain conditions hold for left ideals. T h e proof of [19, Theorem 5, p. 34] remains valid for the principal left ideal domain R. Hence /? is a U F D .

LEMMA 12. Let R be the polynomial ring satisfying all the conditions of Theorem 1, then also every right ideal is principal, i.e. R is a non-commutative principal ideal domain.

Proof. This merely depends on the fact that if the monomorphism a has an inverse fi every polynomial in R with left-hand coefficients can always be represented as a polynomial with right-hand coefficients.

Yvomy-a = {afi)y — afih (cf. (44)) andy^a = {afi'')y'^ (cf. (46)) we conclude

{phy =

0.

Using (44) we obtain successively ax = x{ya)x = x{afi) —x{afih)x =

= x{aP) -x^{afihfi) +x^{afihfih)x = . . . =

T h e last term becomes zero and we have the right-hand commutation formula

ax = xaP-x2a{ph)P-^x^a{fihyp+...+x'a{-ph)'-'^P, (136) and instead of the nilpotent a-derivation h we have the derivation 5 = —fih

again nilpotent of index r, satisfying

{ab)è = {a6)b + {afi){bè) for all a, b e K,

in other words, 6 is a (1, /J)-derivation of the field K. If we apply the methods of Lemma 10 to the right-hand polynomials, then Lemma 12 follows. Summing up we have

THEOREM 8. Let R = K[x] be the ring of skew polynomials in an indeterminate x over a {non-commutative) field K with an cndomorphism a: a^ ai, multiplication in R being defined by

x-a = aix-^a2X^-\-...-\-arxr, at G K.

If we assume that the mappings hti a-> ai {i = 2, 3, ..., r) are right K-independent we can prove

(z) R is a principal left ideal domain with a right Euclidean algorithm; {ii) R is a unique factorization domain;

CHAPTER IV

S T R U C T U R E O F A F I E L D K W I T H A N I L P O T E N T a - D E R I V A T I O N h

§ 13 T h e q u a d r a t i c c a s e

Let K he a (skew) field with an automorphism a (inverse fi) and a nilpotent a-derivation h of index 2, which satisfies

{ah)h = {ah){ha)+a{bh) for all a, b G K. (137) From

0 = {ab)h2 = ah-b{ah + ha) we obtain just as in Chapter I the relations

aó + (5a = 0, (138) Ph+hp = 0. (139) Throughout this section z' will be an element of K such that ih = 1. Greek

letters, except a, fi and h, will denote constants. The field C = {aGK I ah = 0}

is said to be the constant field of (5.

If we choose b — —ifi in (137) we obtain by (139)

a = —{a{ifi)}h-\-{ah)i = TT + ^Z' (JT, Q G C) for all a e K.

If we choose a == z' in (137) and replace b by b^ we obtain in the same way b = {i{bp)}h + i{bdp) = a+ir {a, r G C) for all b G K.

This proves

LEMMA 13. Let K be afield with an automorphism a. Let the a-derivation h of K he nilpotent of index 2, then KjC is a left and right quadratic extension of the constant field C with basis 1 and i {ih = 1).

These quadratic extensions were described by COHN [9]. He proved that the free product of two quadratic extensions over the common ground field can be considered as a skew polynomial ring with a quadratic commutation formula. We shall try to prove the converse. To apply some of his results we shall use his notations. Only we shall deal with left extensions instead of right exten-sions. Of course we can express z' • Q and z'2 in the left C-basis (1 ,z'):

i.Q = {QS)i+eD, all QGC, (140) z'2 + Az'+/z = 0, where 2, ft e C. (141)

It is easy to verify that the mapping S is an cndomorphism of C, and D a (1,5')-derivation of C which satisfies

{ar)D = {aD)T+{aS){rD) {a, TGC). (142) Differentiating (140) we obtain by (137)

ga = QS, all Q G C. (143)

Differentiating (141) we get

z'a = - z ' - A , (144) za2 = z + (A-Aa). (145) It now appears possible to prove

THEOREM 9. Let R = K[x] be the ring of polynomials in x over a {skew) field K with an automorphism a {inverse fi), multiplication in R being defined by

x-zz = ((za)x+(aó«)x2, a G K, (146) so that the a-derivation h is nilpotent of index 2. Then the polynomial ring R is the free

product over the constant field C of the fields K and Ki = xKx^'^ = x~^Kx.

Proof. By Theorem 6, (iv) we know already that R is a homomorphic image of the free product of K and Ki. We only prove that the basic quadratic com-mutation formula (146) follows from the product rules in K and Ki. For further calculations we refer to [9], p. 548 and p. 551. First we show that the defining equations oï K and Ki are the same. We derive from (141)

(z'/?)2 + (A|3)(z'^) + (/z^) = 0 , hence with z'o = x{ifi)y we obtain

fo2+Az'o+/.z = 0, (147) (observe that x{Xfi)y = k and x{/ifi)y = // by (146)).

From (140) we derive in the same way

{ifi){efi) = {QSfi){ifi)+eDfi,

so

x{ifi)y-Q = QS-x{ifi)y-\~QD, or

io-Q = {QS)iü^QD (148) and from (147) and (148) we see that K and Ki have the same defining

equa-tions.

If we put t = z —z'o we obtain the commutation formula for / in a few lines. Formula (147) gives