Hilbert's Theorem 90

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In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that is

then there exists in L such that

The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht (Hilbert 1897, 1998), although it is originally due to Kummer (1855, p.213, 1861).

Often a more general theorem due to Emmy Noether (1933) is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial:

Examples

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Let   be the quadratic extension  . The Galois group is cyclic of order 2, its generator   acting via conjugation:

 

An element   in   has norm  . An element of norm one thus corresponds to a rational solution of the equation   or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every such element a of norm one can be written as

 

where   is as in the conclusion of the theorem, and c and d are both integers. This may be viewed as a rational parametrization of the rational points on the unit circle. Rational points   on the unit circle   correspond to Pythagorean triples, i.e. triples   of integers satisfying  .

Cohomology

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The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of any (not necessarily finite) Galois extension L of a field K with corresponding Galois group G, then

 

Specifically, group cohomology is the cohomology of the complex whose i-cochains are arbitrary functions from i-tuples of group elements to the multiplicative coefficient group,  , with differentials   defined in dimensions   by:

 

where   denotes the image of the  -module element   under the action of the group element  . Note that in the first of these we have identified a 0-cochain  , with its unique image value  . The triviality of the first cohomology group is then equivalent to the 1-cocycles   being equal to the 1-coboundaries  , viz.:

 

For cyclic  , a 1-cocycle is determined by  , with   and:

 

On the other hand, a 1-coboundary is determined by  . Equating these gives the original version of the Theorem.


A further generalization is to cohomology with non-abelian coefficients: that if H is either the general or special linear group over L, including  , then

 

Another generalization is to a scheme X:

 

where   is the group of isomorphism classes of locally free sheaves of  -modules of rank 1 for the Zariski topology, and   is the sheaf defined by the affine line without the origin considered as a group under multiplication. [1]

There is yet another generalization to Milnor K-theory which plays a role in Voevodsky's proof of the Milnor conjecture.

Proof

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Let   be cyclic of degree   and   generate  . Pick any   of norm

 

By clearing denominators, solving   is the same as showing that   has   as an eigenvalue. We extend this to a map of  -vector spaces via

 

The primitive element theorem gives   for some  . Since   has minimal polynomial

 

we can identify

 

via

 

Here we wrote the second factor as a  -polynomial in  .

Under this identification, our map becomes

 

That is to say under this map

 

  is an eigenvector with eigenvalue   iff   has norm  .

References

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  1. ^ Milne, James S. (2013). "Lectures on Etale Cohomology (v2.21)" (PDF). p. 80.
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