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:

Examples

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 c, d) 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

Cohomology

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.

Proof

Elementary

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 .

References

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