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Hilbert's Theorem 90

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 a cyclic extension of fields with Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, 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 famous 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 Galois group G = Gal(L/K), then the first cohomology group is trivial:

Contents

ExamplesEdit

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

 

An element   in L has norm  . An element of norm one 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 y of norm one can be parametrized (with integral cd) as

 

which 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  

CohomologyEdit

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

 

A further generalization using non-abelian group cohomology states that if H is either the general or special linear group over L, then

 

This is a generalization since   Another generalization is

 

for X a scheme, and another one to Milnor K-theory plays a role in Voevodsky's proof of the Milnor conjecture.

ProofEdit

ElementaryEdit

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

 

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

 

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

 

we identify

 

via

 

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

Under this identification, our map

 

That is to say under this map

 

  is an eigenvector with eigenvalue   iff   has norm  .

ReferencesEdit