On the ramification set of a positive quadratic form over an algebraic number field

LXVI.2 (1994)

On the ramification set of a positive quadratic form over an algebraic number field

by

Martin Epkenhans (Paderborn)

1. Introduction and notation. Let A be a finite-dimensional com- mutative and ´etale algebra over K, i.e. a finite product of separable and finite field extensions of K. With it we associate the trace form which is the following non-degenerate quadratic form over K:

A → K, x 7→ tr

(x

) .

It is denoted by hAi. By a quadratic form over K we always mean a non- degenerate quadratic form. We know that a quadratic form ψ over an alge- braic number field K of dimension m ≥ 4 is isometric to a trace form of a field extension of K if and only if the signatures of ψ are non-negative for all real orderings of K (see [9]). Following P. E. Conner and R. Perlis [4] we call a Witt class X of the Witt ring W (K) algebraic if X contains a trace form of a field extension of K. Let K be an algebraic number field. The ramification set Ram(X) of an algebraic Witt class X consists of those finite spots p of K which are ramified in every field extension L/K with hLi ∈ X ([4], p. 166).

Let p be a finite spot of K and let κ

be the residue class field of K at p.

Consider the second residue class homomorphism ∂

: W (K) → W (κ

) (see [22], 6.2.5). The investigation of trace forms over local fields gives ∂

hLi = 0 for all finite spots p of K which are unramified in L/K. In [5] P. E. Conner and N. Yui conjectured that for an algebraic class X ∈ W (Q) we get

Ram(X) = {p | p is finite and ∂

X 6= 0} .

Our main result implies the validity of this conjecture. Let Ω

be the set of spots of K.

Definition 1. Let ψ be a quadratic form over the algebraic number field K with non-negative signatures. The ramification set Ram(ψ) of ψ is defined by

Ram(ψ) = {p ∈ Ω

| p is finite and p is ramified

in every extension L/K with ψ '

hLi} .

Here '

denotes the isometry of quadratic forms over K. We call ψ a positive form if all signatures of ψ are non-negative. In this paper we determine the ramification set of a positive form. In particular, we prove the following. Let ψ be a quadratic form with non-negative signatures and let T ⊂ Ω

be a finite set of finite spots with T ∩ Ram(ψ) = ∅. Then there is a field extension L/K with ψ '

hLi and all p ∈ T are unramified in L/K.

The proof of this result is organized as follows. We start with forms of dimension n = 4. Next suppose n = 2

≥ 8. Then we can choose a quadratic field extension F/K such that all p ∈ T are quadratically unramified in F/K and such that there is a positive form ϕ over F with ψ '

tr

(ϕ) and {P ∈ Ω

| P ∩ o

= p ∈ T } ∩ Ram(ϕ) = ∅. Hence by induction we get the result for forms of dimension 2

. As usual, we write P | p to indicate that P ∈ Ω

is a spot lying above p ∈ Ω

, and tr

(ϕ) is the “Scharlau transfer” of the form ϕ (see [22], p. 47). We treat forms of arbitrary even dimension in a similar way. Next we consider forms of odd dimension. We use Mestre’s deformation process. We can choose trace forms ψ

of dimension 1, 2 or 4 with Ram(ψ

) ∩ T = ∅ and ψ '

⊥ψ

. Hence ψ is isometric to the trace form of some ´etale algebra A = K

× . . . × K

and all p ∈ T are unramified in every field extension K

/K. Then we prove that there is a deformation of the algebra A leaving the trace form intact and preserving the decomposition structure of all spots p ∈ T .

We call ψ a normal (abelian, cyclic) trace form if there is a normal (abelian, cyclic) field extension L/K with ψ '

hLi. In [7] we determined all normal (abelian, cyclic) trace forms of an algebraic number field. In this paper we investigate the Galois ramification set GRam(ψ) of a normal trace form ψ, i.e. the set of all finite spots which are ramified in every Galois extension L/K with ψ '

hLi. In general, Ram(ψ) and GRam(ψ) coincide if ψ is a normal trace form.

We begin by fixing our notations. Let K be an algebraic number field.

Then o

is the ring of integers of K. Let p ∈ Ω

be a spot. Then K

is a completion of K at p. If p is a finite spot, then v

: K → Z denotes the normalized valuation of K defined by p. ∆

∈ o

is an element which is a non-square unit at p such that K

( p

∆

)/K

is unramified. Let L/K be a finite field extension and let p ∈ Ω

, P ∈ Ω

be spots with P | p. The inertia degree of P | p is denoted f (P/p). If L/K is a Galois extension, then we also write f

(L/K) and we set n

(L/K) = [L

: K

]. If L/K is any finite field extension, then Λ

= N

(L

) · K

.

ha

, . . . , a

i denotes the diagonal form a

t

+ . . . + a

t

. Let ψ be a

quadratic form over K. Then dim

ψ is the dimension of ψ, det

ψ ∈ K

is its determinant. Let a, b ∈ K

. Then (a, b)

denotes the generalized

quaternion algebra generated over K by i, j and satisfying i

= a, j

= b,

ij = −ji. The class of (a, b)

in the Brauer group Br(K) of K is also denoted by (a, b)

. Let ψ '

ha

, . . . , a

i be a diagonalization of ψ. The Hasse invariant H

ψ of ψ is defined by

H

ψ = O

i<j

(a

, a

)

∈ Br(K) .

I

(K) is the nth power of the fundamental ideal of W (K). The Scharlau transfer of the one-dimensional form hλi, λ ∈ L

, is called the scaled trace form, and denoted by hLi

.

Again, let K be an algebraic number field and let p ∈ Ω

be a finite spot. H

ψ ∈ {−1, 1} is the local Hasse invariant of ψ and (a, b)

denotes the local Hilbert symbol. Let p ∈ Ω

be a real spot. Then sign

ψ denotes the signature of ψ with respect to the ordering induced by p.

2. The main results. We get the following well-known result from local trace form considerations (see [4], I.5, II.5 or [11]). This gives the necessary condition for p 6∈ Ram(ψ). For the convenience of the reader we sketch a proof.

Proposition 1. Let L/K be a finite extension of algebraic number fields.

(1) Let p ∈ Ω

be a finite spot. If p is unramified in L/K, then p is unramified in K( p

det

hLi)/K and H

hLi = (2, det

hLi)

.

(2) Let p ∈ Ω

be a real spot. Then [L

: K

] = 1 for all spots P ∈ Ω

lying above p if and only if sign

hLi = [L : K].

P r o o f. (1) Let p ∈ Ω

. We know

hL ⊗

K

i '

⊥

hL

i

(see [4], I.5.1). If p is unramified in L/K, then the local extension L

/K

is unramified for any P ∈ Ω

lying above p. The trace form of an unramified local extension is first determined in [11]. Let L

/K

be an unramified local extension of degree f . Then hL

i '

f · h1i if f is odd.

Let f be even. Then K

( p

det

hL

i)/K

is the unique unramified extension of degree 2 and we get H

hL

i = (2, det

hL

i)

(see also [8], Theorem 1).

(2) Let p be a real spot of K. By a classical result of Sylvester we know that the signature sign

hLi equals the number of spots P ∈ Ω

lying above p and such that the local degree is 1 ([22], 3.2.6 or [23]).

We now state the main results of this paper.

Theorem 1. Let K be an algebraic number field. Let ψ be a positive quadratic form over K of dimension ≥ 4 or let ψ '

h2, 2Di, D 6∈ K

or ψ '

h1, 2, Di, D ∈ K

. Let T ⊂ Ω

be a finite set of finite spots with p unramified in K( √

det

ψ)/K and H

ψ = (2, det

ψ)

. Then there

is a field extension L/K with ψ '

hLi and such that all spots p ∈ T are unramified in L/K. If n = dim

ψ is even, we can choose L/K such that f (P/p) ∈ {n, n/2} for all P ∩ o

= p ∈ T . In particular ,

Ram(ψ)

= {p ∈ Ω

| p ramifies in K( p

det

ψ)/K or H

ψ 6= (2, det

ψ)

} . Corollary 1. The conjecture of Conner and Yui holds true.

P r o o f. If p is a non-dyadic spot, then p is unramified in K( p

det

hLi)/K and H

hLi = (2, det

hLi)

is equivalent to ∂

hLi = 0. Hence by Theo- rem 1 it remains to consider the prime 2. Let X ∈ W (Q) be a Witt class with ∂

X = 0, i.e. ord

dis X is even. Choose a quadratic form ψ ∈ X such that det

ψ ≡ 1, 5 mod 8. Then H

ψ 6= H

(ψ ⊥ 2 · h1, −1i) and ψ ⊥ 2 · h1, −1i ∈ X. Hence we can choose ψ ∈ X such that 2 is unram- ified in Q( p

det

ψ) and H

ψ = (2, det

ψ)

. By Theorem 1 there is a field extension L/Q such that 2 is unramified in L/Q and hLi '

ψ ∈ X.

In the next theorem we consider the Galois ramification set of a normal trace form. Let µ

be the group of dth roots of unity.

Theorem 2. Let K be an algebraic number field and let ψ be a quadratic form of dimension n = 2

m, m odd, over K. Let D ∈ K

with det

ψ ≡ D mod K

.

(1) Let n be odd. Then ψ is a normal trace form iff ψ is a cyclic trace form iff ψ '

n · h1i. Then GRam(ψ) = Ram(ψ) = ∅.

(2) Let n = 2m ≡ 2 mod 4. Then ψ is a normal trace form iff ψ is a cyclic trace form iff ψ '

m · h2, 2Di and D 6∈ K

. Then GRam(ψ) = Ram(ψ).

(3) Let n = 2

m ≡ 0 mod 4 and D ∈ K

. Then ψ is a normal trace form iff ψ is an abelian trace form iff sign

ψ ∈ {0, n} for all real spots p ∈ Ω

. But ψ is not a cyclic trace form. Then GRam(ψ) = Ram(ψ) = {p ∈ Ω

| H

ψ = −1}.

(4) Let n = 4m ≡ 4 mod 8 and D 6∈ K

. Then ψ is a normal trace form iff ψ is a cyclic trace form iff D = a

+ b

with a, b ∈ K and ψ '

m · h1, D, c, ci for some c ∈ K

. Then GRam(ψ) = Ram(ψ).

(5) Let n = 2

m ≡ 0 mod 8 and D 6∈ K

. Then ψ is a normal trace form iff ψ is a cyclic trace form iff H

ψ = (2, D)

, sign

ψ ∈ {0, n} for all real spots p ∈ Ω

and K( √

D)/K is contained in a cyclic extension of degree 2

. Then H

ψ = 1 for all non-dyadic spots and for all infinite spots p ∈ Ω

. Set T

:= {p ∈ Ω

, p | 2 and p is completely non-split in K(µ

)/K}. Then either GRam(ψ) = Ram(ψ), or K(µ

)/K is not cyclic, T

= {p

} and p

is unramified in K( √

D)/K. Then Ram(ψ) ⊂ GRam(ψ) ⊂ Ram(ψ) ∪ {p

}.

(6) Let ψ be a normal (cyclic) trace form and let T ⊂ Ω

be a finite set

of finite spots with T ∩ GRam(ψ) = ∅. If det

ψ 6∈ K

and n ≡ 0 mod 8

suppose K(µ

)/K is cyclic or T

= ∅ or T

6⊂ T . Then there is an abelian (cyclic) field extension L/K with ψ '

hLi and all p ∈ T are unramified in L/K.

3. Positive forms of even dimension. We start with forms of dimen- sion 4.

Lemma 1. Let K be a field of char(K) 6= 2 and let f (X) = X

− 2aX

+ b ∈ K[X] be an irreducible and separable polynomial. Set L = K[X]/(f (X)).

Then

hLi '

h1, a

− b, ab, a(a

− b)i . Hence det

hLi ≡ b mod K

and

H

hLi = (a, −b(a

− b))

⊗ ((a

− b), −1)

. For a proof see [4], Theorem I.10.1.

Lemma 2. Theorem 1 holds for quadratic forms of dimension 4.

P r o o f. Let ψ '

h1, u, v, uvDi and suppose T 6= ∅. The equation ux

+ vx

+ uvDx

= D(Dx

− 1)

has a solution in K since ψ is a positive form of dimension 4. The set S = {p ∈ Ω

| p is real or p 6∈ T with H

ψ 6= (−D, −1)

} is finite and disjoint from T . Let τ ∈ K be an element with

(1) (Dτ

− 1)(Dx

− 1) ∈ K

for all p ∈ S and

(2) D(Dτ

− 1)∆ ∈ K

for all p ∈ T , where ∆ ∈ K

is a non-square unit at all p ∈ T such that K

( p

∆

)/K

is unramified.

Set g(X) = X

+ 2Dτ

X + Dτ

. The discriminant of g(X) satisfies dis(g(X)) = 4Dτ

(Dτ

− 1) ≡ ∆ mod K

for all p ∈ T . Since T 6= ∅, the polynomial g(X) is irreducible. Let g(β) = 0 and set F = K(β). By the Hasse–Minkowski Local Global Principle hu, v, uvDi represents D(Dτ

− 1).

We can choose w ∈ K

such that −wβ 6∈ F

and

ψ '

h1, D(Dτ

− 1), w, w(Dτ

− 1)i . Hence

h(X) = g(−X

w

)w

= X

− 2Dwτ

X

+ Dw

τ

∈ K[X]

is irreducible. Set M = K[X]/(h(X)). From Lemma 1 we know ψ '

hM i.

The extension K

( p

D(Dτ

− 1))/K

is quadratically unramified for all p ∈ T since D(Dτ

− 1)∆ ∈ K

.

Now let p ∈ T be a non-dyadic spot which ramifies in M/K, hence p = P

with f (P/p) = 2. From [6], Satz 5.5(3), we know 1 = H

hM i =

−(π, −D)

, where π ∈ K is a prime at p. Hence −D∆ ∈ K

for these

spots, which gives −(Dτ

−1) ∈ K

. Therefore the form h1, (Dτ

− 1)i '

h1, −1i is isotropic over K

.

Next let p ∈ T be a dyadic spot and let P ∈ Ω

, e P ∈ Ω

be spots with e P | P and P | p. Suppose that D ∈ K

and −wβ 6∈ F

. Then Me

= K

( √

∆, √

z) for some z ∈ K

with −wβz ∈ F

. We further get 1 = H

ψ = H

hM i = (−∆, z)

. Thus h1, Dτ

− 1i '

h1, ∆i represents z ∈ K

over K

.

Let p ∈ T be a dyadic spot with D∆ ∈ K

. Then N

(−wβ) ≡ N

(β) = g(0) ≡ ∆ mod K

. Thus [Me

: K

] = 4 and Me

/K

is a cyclic extension (see [6], Satz 2.2(3)(b)). Every square class in the kernel of the map N

: F

→ K

contains some z ∈ K

. Hence there is some z ∈ K

with −wβz ≡ ∆

mod F

. We get

(z, −1)

= H

hF

i

· H

hF

i

= H

hF

i

· H

hF

i

= H

(tr

(h2, 2∆

i)) · H

hMe

i = 1 , since M = F ( √

−wβ) and tr

(h2, 2∆

i) is the trace form of the unique unramified extension of K

having degree 4. Thus h1, Dτ

− 1i '

h1, 1i represents z ∈ K

over K

.

Hence h1, Dτ

− 1i represents some z ∈ K

with

(1) v

(z) is odd, if p ∈ T is a non-dyadic spot which ramifies in M/K, (2) z ∈ K

if p ∈ T is unramified in M/K,

(3) −wβz ∈ F

, if p is a dyadic spot with D ∈ K

, −wβ 6∈ F

and p ramifies in M/K and

(4) F

( √

−wβz )/K

is unramified of degree 4 if D∆ ∈ K

and p ∈ T is a dyadic spot which ramifies in M/K.

Hence hw, w(Dτ

− 1)i '

hzw, zw(Dτ

− 1)i. Now f (X) = X

− 2Dwzτ

X

+ Dw

z

τ

∈ K[X]

defines the desired field extension of degree 4.

Let F/K be a finite field extension of algebraic number fields with [F : K] = m. M. Kr¨ uskemper [16] investigated the transfer of quadratic forms. He gave sufficient conditions for a positive quadratic form ψ of dimen- sion nm, n ≥ 3, to be the transfer of a positive form ϕ over F . The proof of the next result follows the lines of [16], Lemma 7, and [15], Lemma 1, where a similar result is proven without taking care of the ramification of primes.

We construct the field extensions with the help of Grunwald’s Theorem.

This simplifies some of Kr¨ uskemper’s original proofs.

Proposition 2. Let ψ be a positive quadratic form of dimension mn,

n ≡ 0 mod 4, over the algebraic number field K. Let T ⊂ Ω

be a finite set

of finite spots which are unramified in K( √

det

ψ)/K and such that H

ψ = (2, det

ψ)

. Let F/K be a Galois extension of degree m with det

ψ ∈ Λ

, sign

hF i = m for all real spots p ∈ Ω

and f

(F/K) = m for all p ∈ T . Suppose, further , that T contains no dyadic spot if m is even and m 6= 2. Then there is a positive quadratic form ϕ over F with

(1) ψ '

tr

(ϕ) and

(2) all P ∈ Ω

with P ∩ o

= p ∈ T are unramified in F ( √

det

ϕ)/F and we get H

ϕ = (2, det

ϕ)

for these spots.

P r o o f. Let T

be the set of spots P of F for which P ∩ o

= p ∈ T . First let ϕ be an arbitrary quadratic form of even dimension over F such that all P ∈ T

are unramified in F ( √

det

ϕ)/F . A manipulation with Hasse invariants and Hilbert symbols implies H

ψ = H

ϕ for all p ∈ T , P ∈ T

, P ∩ o

= p (use [6], Satz 0.6 and Satz 3.9). Thus we only have to prove that there is a positive form ϕ over F with ψ '

tr

(ϕ) and all P ∈ T

are unramified in F ( √

det

ϕ)/F .

First let ψ be a torsion form. Let m be even. Then det

ψ is totally positive, hence a sum of squares. Since F/K is a Galois extension with det

ψ ∈ Λ

, there is a totally positive element λ

∈ F with N

(λ

) ≡ det

ψ mod K

(apply [16], Proposition 7(b)). We can choose some totally positive z ∈ K such that zλ

∆

∈ F

or zλ

∈ F

for all P ∈ T

. Note that T contains no dyadic spots if m 6= 2. Set λ = z · λ

. If m is odd, set λ = det

ψ. Hence N

(λ) ≡ det

ψ mod K

, λ is totally positive and all P ∈ T

are unramified in F ( √

λ)/F . Now ψ − tr

(h1, −λi) is a torsion form in I

(K). By a result of Leep and Wadsworth (see [18], Theorem 1.11, resp. [14], Theorem 1.2) there is a torsion form % ∈ I

(F ) with

tr

(%) = ψ − tr

(h1, −λi) .

Of course, the torsion form % ⊥ h1, −λi is a positive form with det

(% ⊥ h1, −λi) ≡ λ ≡ 1, ∆

mod F

for all P ∈ T

.

Finally, let ψ be an arbitrary form for which the condition of the propo- sition holds. We can choose a form % over F such that

(1) dim

% ≡ 0 mod 4,

(2) 0 ≤ sign

% ≤ n for all real spots P ∈ Ω

, (3) P

P|p

sign

% = sign

ψ for all real spots p ∈ Ω

, (4) all P ∈ T

are unramified in F ( √

det

%)/F and H

% = (2, det

%)

, where det

% ∈ F

iff det

ψ ∈ K

, P | p.

Thus by [22], 3.4.5, tr

(%) − ψ is a torsion form with

det

(tr

(%) − ψ) ≡ N

(det

%) · det

ψ ≡ 1 mod K

and

H

(tr

(%) − ψ) = H

(tr

(%)) · H

(−ψ) · (N

(det

%), det

ψ)

= H

% · H

ψ = (2, det

%)

· (2, det

ψ)

= 1

for all p ∈ T . We get det

(tr

(%) − ψ) ∈ Λ

. By the above, there is a torsion form τ over F with tr

(τ ) = tr

(%) − ψ and all P ∈ T

are unramified in F ( √

det

τ )/F and H

τ = (2, det

τ )

for these spots. The Witt class of % − τ can be represented by a form ϕ of dimension n (see [22], 6.6.6). Hence ψ '

tr

(ϕ). It follows that det

ϕ ≡ det

% · det

τ mod F

and H

ϕ = H

(% − τ ) = H

% · H

(−τ ) · (det

%, det

(−τ ))

= (2, det

ϕ)

for all P ∈ T

.

Lemma 3. Theorem 1 holds for positive forms of dimension n = 2

m, m odd, l ≥ 2.

P r o o f. First let m = 1. We proceed by induction on l. If l = 2, use Lemma 2. Let l ≥ 3. We can choose some totally positive P ∈ o

such that

(1) K

( √

P )/K

is quadratic unramified for all p ∈ T and (2) (det

ψ, P )

= 0, hence det

ψ ∈ Λ

, where F = K( √

P ).

Now apply Proposition 2. Next let m be an arbitrary odd number. By the Theorem of Grunwald–Hasse–Wang [21], Korollar 6.9, we can choose some cyclic field extension F/K of degree m with f

(F/K) = m for all p ∈ T . Now apply Proposition 2 again.

We have to consider forms of dimension n ≡ 2 mod 4 separately since forms of dimension 2 are somewhat exceptional. A binary quadratic form ψ is a trace form iff ψ '

h2, 2Di with D 6∈ K

. Based on a result of E. Bender [1], M. Kr¨ uskemper gave a local global principle for scaled trace forms of odd dimension over an algebraic number field (see [16], Theorem 1).

We give a stronger version of this result.

Proposition 3. Let F/K be an extension of algebraic number fields of odd degree m. Let ψ be a quadratic form of dimension m over K with

|sign

ψ| ≤ sign

hF i for all real spots p ∈ Ω

and H

ψ = (det

ψ, −1)

for all non-dyadic spots p ∈ Ω

for which there is only one spot of F lying above p. Let S be a finite set of finite spots containing all dyadic spots and all non-dyadic spots which ramify in F/K or in K( √

det

ψ)/K or for which H

ψ 6= (det

ψ, −1)

.

(1) Then for every p ∈ S there is some λ

∈ F

with hF i

'

ψ.

(2) Suppose that for every p ∈ S there is some λ

∈ K

with hF i

'

ψ. Then there is some λ ∈ F

with ψ '

hF i

and λ · λ

∈ F

for all

P ∈ Ω

with P ∩ o

= p ∈ S.

P r o o f. (1) Let F

/K

be an extension of non-dyadic fields with [F

: K

] = m. Then H

hF

i

= (det

hF

i

, −1)

(see [6], Satz 4.2). If p is a finite spot which splits over F , then use [16], Lemma 6. If p does not split over F , apply [2], Lemma 3, for dyadic spots, and note that hF i

, λ = det

hF i · det

ψ and ψ have the same determinant.

(2) See [16], Proofs of Proposition 1 and Theorem 1.

Lemma 4. Theorem 1 holds for quadratic forms ψ of dimension n = 2m ≡ 2 mod 4.

P r o o f. By the above we can assume n 6= 2. Assume further T 6= ∅.

Choose some non-dyadic spot p

∈ Ω

with ∂

ψ = 0, p

6∈ T and −1 ∈ K

. Let S ⊂ Ω

be the set of non-dyadic spots with ∂

ψ 6= 0. Then S is a finite set with S ∩ T = ∅. Because of the Theorem of Grunwald–Hasse–

Wang [21], Korollar 6.9, there is a cyclic field extension F/K of degree m such that

(1) f

(F/K) = m for all p ∈ T , (2) n

(F/K) 6= m for all p ∈ S, (3) n

(F/K) = 1.

Let T

be the set of spots P of F with P ∩ o

= p ∈ T . Then sign

(h2i ⊗ ψ − hF i) = sign

ψ − m and sign

ψ ≥ 0 gives |sign

(h2i ⊗ ψ − hF i)| ≤ m for all real p ∈ Ω

. Therefore there is a form ϕ of dimension m over K which is Witt-equivalent to h2i ⊗ ψ − hF i (see [22], 6.6.6). Thus ψ '

h2i ⊗ ϕ ⊥ hF i

. Let p be a non-dyadic spot with (det

ϕ, −1)

6= H

ϕ. Then v

(det

ϕ) ≡ 1 mod 2 or H

ϕ = −1. We know H

ϕ = H

(h2i ⊗ ψ) = (2, det

ψ)

·H

ψ. Hence p ∈ S. But n

(F/K) 6= m for these spots. Therefore we can choose some λ

∈ F

with hF i

'

ϕ for these spots.

Let p ∈ T . Set λ

= 1 if det

ψ ∈ K

and λ

= ∆

if det

ψ 6∈ K

, where P ∈ T

is the unique spot lying above p ∈ T . Then hF i

'

ϕ (use [6], Satz 3.9(3)).

Fix some P

∈ Ω

lying above p

. If ψ '

n · h1i, let a ∈ F

be an element with v

(a) ≡ 1 mod 2 and a ∈ F

for all P 6= P

, P | p

. Set λ

= a · σ(a) with hσi = G(F/K). Then λ

6∈ F

and hF i

'

m · h1i (see proof of [15], Proposition 2).

By Proposition 3 there is some λ ∈ F

with ϕ '

hF i

and λ ∈ F

if det

ψ ∈ K

, resp. λ · ∆

∈ F

if det

ψ 6∈ K

for all p ∈ T and λ · λ

∈ F

if ψ '

n · h1i. Thus ψ '

tr

(h2, 2λi).

Now λ ∈ F

gives ψ '

tr

(h2, 2i) '

n · h1i, which contradicts λ ≡ λ

6≡ 1 mod F

. Set L = F ( √

λ). Then all P ∈ T

are unramified in

L/F .

4. Positive forms of odd dimension. Now we treat positive forms of odd dimension. We modify our original proof of [9], resp. [10], which is based on a deformation process of Mestre [20]. We first recall this result.

Proposition 4. Let K be an algebraic number field. Let f

(X), . . . , f

(X)

∈ o

[X] be monic polynomials such that f (X) = f

(X) . . . f

(X) has odd degree m ≥ 3. Then there are monic polynomials p

(X), . . . , p

(X) ∈ o

[X]

and a polynomial q(X) ∈ o

[X] such that

(1) K[X]/(f

(X)) ' K[X]/(p

(X)) for i = 1, . . . , s.

(2) deg(q(X)) < deg(f (X)).

(3) p(X) = p

(X) . . . p

(X) and q(X) are relatively prime. Hence F (T, X) = p(X) − T q(X) ∈ o

[T, X]

is irreducible.

(4) For every τ ∈ K the trace forms of K[X]/(f (X)) and K[X]/(F (τ, X)) over K are isometric.

P r o o f. See [20], Proposition (1) and (2), and [9], Theorem 2.

We need this result in the following version.

Proposition 5. Let K be an algebraic number field. Let f (X) ∈ K[X]

be a monic separable polynomial of odd degree m ≥ 3. Let T ⊂ Ω

be a finite set of finite spots. There is a polynomial F (T, X) ∈ o

[T, X] and there are infinitely many elements τ ∈ K such that F (τ, X) ∈ o

[X] is a monic irreducible polynomial with the following properties:

(1) hK[X]/(f (X))i '

hK[X]/(F (τ, X))i.

(2) Let p ∈ T and let f (X) = f

(X) . . . f

(X) be the decomposition of f (X) into monic prime factors in K

[X]. Then F (τ, X) factors in K

[X]

as F (τ, X) = F

(X) . . . F

(X) and

K

[X]/(f

(X)) ' K

[X]/(F

(X)) for i = 1, . . . , r ;

i.e. f (X) and F (τ, X) have the same ramification structure for all p ∈ T . P r o o f. Let p(X), q(X) ∈ o

[X] be as in Proposition 4. Obviously, the ramification structures of f (X) and of p(X) coincide for all spots p ∈ Ω

. Let π ∈ o

be an element with v

(π) > 0 for all p ∈ T . We can choose some s ∈ N such that F (π

T, X) has the following property.

For every τ ∈ o

the polynomials F (π

τ, X) and p(X) (hence F (π

τ, X) and f (X)) have the same ramification structure for all p ∈ T . Use [3], IV § 3 Satz 1 and Bemerkung and [17], Proposition 4, or apply [19], Exercise 24.22.

Then use Hilbert’s Irreducibility Theorem.

Lemma 5. Let K be an algebraic number field. Let ψ be a positive

quadratic form of dimension m ≥ 5 over K. Let T ⊂ Ω

be a finite set

of finite spots of K. There are elements a

, . . . , a

∈ o

, s = [(m − 5)/2], and there is a positive quadratic form ϕ of dimension 4 over K such that

(1) ψ '

ϕ ⊥ h2, 2a

i ⊥ . . . ⊥ h2, 2a

i if m is even and (2) ψ '

ϕ ⊥ h1i ⊥ h2, 2a

i ⊥ . . . ⊥ h2, 2a

i if m is odd,

and for all p ∈ T we get a

∈ K

, det

ψ · det

ϕ ∈ K

and H

ψ = H

ϕ.

P r o o f. If m is odd, then ψ '

ψ ⊥ h1i with some positive form e e ψ.

Hence assume m is even. By the Approximation Theorem we can choose some a ∈ K

such that

(1) a ∈ K

if p ∈ T .

(2) Let p ∈ Ω

be a real spot. Then a is negative at p iff sign

ψ 6=

dim

ψ.

The Hasse–Minkowski Local Global Principle gives ψ '

ψ

⊥ hai with sign

ψ

= sign

ψ − sign

a ≥ 0, since dim

ψ ≥ 4. By induction we get ψ '

ψ ⊥ ha e

, . . . , a

i, where a

, . . . , a

∈ K

have the properties (1) and (2). Let p ∈ Ω

be a real spot. Then a

∈ −p implies a

∈ −p for m ≥ j ≥ i.

We further get sign

( e ψ ⊥ s · h−2i) ∈ {0, 2, 4}. Hence e ψ '

ϕ ⊥ s · h2i with dim

ϕ = 4 and ϕ is a positive form.

P r o o f o f T h e o r e m 1. The trace forms of dimension ≤ 3 are h1i, h2, 2Di with D 6∈ K

and h1, 2, Di with D ∈ K

(see [4], III 3.6). By Lemmas 3 and 4 it remains to consider positive forms of odd dimension.

Then use Lemmas 5 and 2 and Proposition 5.

5. Proof of Theorem 2. In [7], Theorem 1, we classified all normal, abelian and cyclic trace forms of an algebraic number field. Hence in view of Proposition 1 we only have to prove (6).

(1) By the Very Weak Existence Theorem of Grunwald [12] there is a cyclic field extension L/K of degree n with n

(L/K) = 1 for all p ∈ T . Hence all p ∈ T are unramified in L/K.

Since the compositum of unramified field extensions is an unramified field extension we can assume n = 2

≥ 2. Hence the proof of (2) is obvious.

(3) Because of the proof of Lemma 2 we can assume n = 2

≥ 8.

(a) ψ ∈ I

(K). By the Theorem of Grunwald–Hasse–Wang [21], Korol- lar 6.9, there is a Galois extension L/K with G(L/K) ' (Z

)

and such that every p ∈ T is unramified in L/K and, for a real spot, n

(L/K) = 2 iff sign

ψ = 0.

(b) ψ 6∈ I

(K). We use the Very Weak Existence Theorem of Grun- wald [12]. There is a cyclic field extension F/K of degree 2

with

(1) n

(F/K) = 2

if H

ψ = −1,

(2) n

(F/K) = 1 if p is a real spot or p ∈ T .

Hence det

hF i 6∈ K

if H

ψ = −1. Thus there is some a ∈ K

with H

ψ = (det

hF i, a)

and a is negative at p iff sign

ψ = 0 (see proof of Proposi- tion 3 in [7]). Choose some totally positive b ∈ K

with (b, det

hF i)

= 0 and ab ∈ K

for all p ∈ T . Set L = F ( √ ab).

(4) Since H

ψ = (c, −1)

, we can define local extensions L(p)/K

as follows (see [6], Satz 3.14, 3.16):

(1) L(p) = C if sign

ψ = 0 and p is a real spot; otherwise let L(p) = R, (2) L(p)/K

is unramified of degree 4 if D∆ ∈ K

and p ∈ T ,

(3) L(p) = K

if D ∈ K

and p ∈ T ,

(4) ψ '

hL(p)i where L(p)/K

is cyclic of degree 4 if D 6∈ K

and either p ∈ Ram(ψ) or p is dyadic,

(5) L(p) = K

( √

c) if D ∈ K

and either p ∈ Ram(ψ) or p is dyadic.

These are finitely many local conditions. By [13], Korollar zu Satz 8, the quadratic extension K( √

D)/K is contained in a cyclic field extension M/K of degree 4 which has the given completion at the above spots, i.e.

M

' L(p) , P | p. Now choose a totally positive element t ∈ K

as follows:

(1) t ∈ K

, if p ∈ T ∪ Ram(ψ) or p is dyadic, (2) v

(t) ≡ 1 mod 2, if H

ψ 6= H

hM i,

(3) v

(t) ≡ 0 mod 2 for all other spots except maybe one non-dyadic spot p

6∈ T ∪ Ram(ψ) with H

ψ = H

hM i.

Set F = K( √

D), M = F ( q

x + y √

D). Then L = F ( q

tx + ty √ D) defines the desired field extension.

(5) Use [13], Korollar zu Satz 8.

References

[1] E. A. B e n d e r, Characteristic polynomials of symmetric matrices, Pacific J. Math.

25 (1968), 433–441.

[2] —, Characteristic polynomials of symmetric matrices ii, Linear and Multilinear Algebra 2 (1974), 55–63.

[3] Z. I. B o r e v i c h and I. R. S h a f a r e v i c h, Zahlentheorie, Birkh¨auser, Basel, 1966.

[4] P. E. C o n n e r and R. P e r l i s, A Survey of Trace Forms of Algebraic Number Fields, World Scientific, Singapore, 1984.

[5] P. E. C o n n e r and N. Y u i, The additive characters of the Witt ring of an algebraic number field, Canad. J. Math. 40 (1988), 546–588.

[6] M. E p k e n h a n s, Spurformen ¨ uber lokalen K¨orpern, Schriftenreihe Math. Inst. Univ.

M¨ unster 44 (1987).

[7] —, Trace forms of normal extensions of algebraic number fields, Linear and Multi- linear Algebra 25 (1989), 309–320.

[8] —, Trace forms of normal extensions of local fields, ibid. 24 (1989), 103–116.

[9] —, On trace forms of algebraic number fields, Arch. Math. (Basel) 60 (1993), 527–

529.

[10] M. E p k e n h a n s and M. K r ¨ u s k e m p e r, On trace forms of ´etale algebras and field extensions, Math. Z., to appear.

[11] V. P. G a l l a g h e r, Local trace forms, Linear and Multilinear Algebra 7 (1979), 167–174.

[12] H. H a s s e, Zum Existenzsatz von Grunwald in der Klassenk¨orpertheorie, J. Reine Angew. Math. 188 (1950), 40–64.

[13] N. K l i n g e n, Das Einbettungsproblem f¨ ur Algebraische Zahlk¨orper bei Beschr¨ankung der Verzweigung, ibid. 334 (1982), 91–115.

[14] M. K r ¨ u s k e m p e r, Algebraic systems of quadratic forms of number fields and func- tion fields, Manuscripta Math. 65 (1989), 225–243.

[15] —, Algebraic number field extensions with prescribed trace form, J. Number Theory 40 (1992), 120–124.

[16] —, On the scaled trace form and the transfer of a number field extension, ibid. 40 (1992), 105–119.

[17] S. L a n g, Algebraic Number Theory, Addison-Wesley, Reading, 1970.

[18] D. B. L e e p and A. R. W a d s w o r t h, The transfer ideal of quadratic forms and a Hasse norm theorem mod squares, Trans. Amer. Math. Soc. 315 (1989), 415–431.

[19] F. L o r e n z, Einf¨ uhrung in die Algebra, Teil II , BI-Wissenschaftsverlag, Mannheim, 1990.

[20] J.-F. M e s t r e, Extensions r´eguli`eres de Q(T ) de groupe de Galois _A e

, J. Algebra 131 (1990), 483–495.

[21] J. N e u k i r c h, ¨ Uber das Einbettungsproblem der algebraischen Zahlentheorie, Invent.

Math. 21 (1973), 59–116.

[22] W. S c h a r l a u, Quadratic and Hermitian Forms, Grundlehren Math. Wiss. 270, Springer, Berlin, 1985.

[23] O. T a u s s k y, The discriminant matrices of an algebraic number field, J. London Math. Soc. 43 (1968), 152–154.

FB MATHEMATIK

UNIVERSIT ¨AT-GESAMTHOCHSCHULE D-33095 PADERBORN, GERMANY E-

mail

Received on 24.2.1993

and in revised form on 20.7.1993 (2387)