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ACTA ARITHMETICA LXXVI.2 (1996)

When are global units norms of units?

by

David Folk (Ypsilanti, Mich.)

1. Introduction. Let L/K be a Galois extension of number fields, ω ∈ K, H the Hilbert class field of L. Often it is possible to determine that ω = N

L/K

(z), for some z ∈ L, while number theoretical questions often demand more: If ω is an element of O

K

, is it also the norm of an element z ∈ O

L

? In this paper, we prove that if ω is not the norm of a unit, then it is not even a global norm from H.

Throughout the paper, we will be using the following notation: For an arbitrary number field M , O

M

will denote the ring of algebraic integers in M , O

M

the group of units of O

M

, P

M

the principal ideals of M . Further, A

M

will denote the adeles of M , A

M

the group of ideles of M , with I

M

denoting the ideals of M . Finally, C

M

and Cl

M

will denote the id`ele class group of M and the ideal class group of M , respectively.

Theorem 1. Let L/K be a Galois extension of number fields with H the Hilbert class field of L. Then

N

H/K

(H

) ∩ O

K

⊂ N

L/K

(O

L

).

P r o o f. Let

G := Gal(L/K) and Γ := Gal(H/K),

with ω ∈ N

H/K

(H

) ∩ O

K

. We will also write ω for its image in the Tate cohomology group b H

0

(Γ, O

H

). This is harmless, since we want to prove that ω ∈ N

L/K

(O

L

) and the other representatives of the class of ω in b H

0

(Γ, O

H

) differ from ω by an element of N

H/K

(O

H

) ⊂ N

L/K

(O

L

). Consider the com- mutative diagram of Γ -modules, with exact rows and exact columns:

1991 Mathematics Subject Classification: 11R37.

[145]

(2)

146 D. Folk

0 0 0

0 O

H

H

P

H

0

0 U

H

A

H

I

H

0

0 U

H

/O

H

C

H

Cl

H

0

0 0 0

²² ²² ²²

// //

²² //

²² //

²² // //

²² //

²² //

²² // //

²² //

²² //

²²

We thus derive the following diagram of Γ -cohomology groups:

H b

−2

(Γ, Cl

H

) H b

−1

(Γ, U

H

/O

H

) H b

−1

(Γ, C

H

)

H b

−1

(Γ, P

H

) H b

0

(Γ, O

H

) H b

0

(Γ, H

)

0 H b

0

(Γ, U

H

) H b

0

(Γ, A

H

)

²² //

²²

δ

// ²²

²² // ²² // ²²

// //

The zero in the lower left corner comes from the fact that b H

−1

(Γ, I

H

) = 0, which follows from Shapiro’s lemma (see [2], Lemma 2.11 for details).

We now do a diagram chase: ω ∈ b H

0

(Γ, O

H

) maps to 0 in b H

0

(Γ, H

).

Therefore it “comes from” z ∈ b H

−1

(Γ, P

H

). Because of the zero in the left

lower corner, z itself “comes from” y ∈ b H

−2

(Γ, Cl

H

). Let x denote the image

of y in b H

−1

(Γ, U

H

/O

H

). Let x ∈ U

H

also denote any unit id`ele whose class

in b H

−1

(Γ, U

H

/O

H

) is our x. By the exactness of the diagram, the class

x maps to 0 in b H

−1

(Γ, C

H

), hence the element x ∈ H

· (A

H

)

IΓ

, where

I

Γ

denotes the augmentation ideal of Z[Γ ], and H

· (A

H

)

IΓ

denotes the

subgroup of A

H

generated by the set {x

1−σ

: x ∈ A

H

, σ ∈ Γ } and H

.

Moreover, by the commutativity of the diagram, the map labeled δ maps

the class x to the class of ω in b H

0

(Γ, O

H

). Recall the definition of the

coboundary map δ : b H

−1

(Γ, U

H

/O

H

) → b H

0

(Γ, O

H

): Choose any element

x ∈ U

H

representing the class x ∈ b H

−1

(Γ, U

H

/O

H

); then δ(x) = the class of

N

H/K

(x) in b H

0

(Γ, O

H

) ∼ = O

K

/N

H/K

(O

H

). Thus we see that the class of the

N

H/K

-norm of the element x in b H

0

(Γ, O

H

) is equal to that of ω. Combining

(3)

When are global units norms of units? 147

this with the fact that b H

−1

(Γ, U

H

/O

H

) is a submodule of U

H

/(U

H

)

IΓ

O

H

, we see that we may modify the element x ∈ U

H

by an element of O

H

so that the elements x and ω satisfy N

H/K

(x) = ω.

Let therefore α ∈ H

such that xα

−1

∈ (A

H

)

IΓ

. Then N

H/L

(xα

−1

) ∈ N

H/L

((A

H

)

IΓ

) = (N

H/L

(A

H

))

IG

⊂ (U

L

· L

)

IG

= U

LIG

· (L

)

IG

.

The first equality follows from the fact that N

H/L

is a central element in the group ring Z[Γ ], the inclusion from the fact that H is the Hilbert class field of L and the last equality is a generality. By I

G

we denote the augmentation ideal of the group ring Z[G].

So, we have N

H/L

(xα

−1

) = uβ, where u ∈ U

LIG

and β ∈ (L

)

IG

. This implies that

η = N

H/L

(x) · u

−1

= N

H/L

(α) · β ∈ L

∩ U

L

= O

L

. Since N

L/K

(u) = 1, we have N

L/K

(η) = N

H/K

(x) = ω, as required.

I would like to express my deep appreciation to David Leep and Ren`e Schoof for their willingness to listen and for their wonderful suggestions.

References

[1] J. W. S. C a s s e l s and A. F r ¨o h l i c h (eds.), Algebraic Number Theory, Academic Press, New York, 1967.

[2] A. F r ¨o h l i c h, Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields, American Mathematical Society, 1980.

Department of Mathematics Eastern Michigan University Ypsilanti, Michigan 48197 U.S.A.

E-mail: mth-folk@online.emich.edu

Received on 24.2.1995

and in revised form on 22.1.1996 (2747)

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