Hilbert's Theorem 90

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In abstract algebra, Hilbert's Theorem 90 (or Satz 90) refers to 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 s and if a is an element of L of relative norm 1, then there exists b in L such that

a = s(b)/b.

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:

H1(G, L×) = {1}

Examples[edit]

Let L/K be the quadratic extension  \mathbb{Q}(i) / \mathbb{Q} . The Galois group is cyclic of order 2, its generator s acting via conjugation:

 s:\, \, a+bi\mapsto a-bi\ .

An element  x=a+bi in L has norm  xx^{s}=a^2 +b^2. An element of norm one corresponds to a rational solution of the equation a^2+b^2=1 or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every element y of norm one can be parametrized (with integral c,d) as

 y={{c+di}\over{c-di}}={{c^2-d^2}\over{c^2+d^2}}+{2dc\over{c^2+d^2}}i

which may be viewed as a rational parametrization of the rational points on the unit circle. Rational points \, (x,y)=(a/c,b/c) on the unit circle  x^2+y^2=1 correspond to Pythagorean triples, i.e. triples \,(a,b,c) of integers satisfying \, a^2+b^2=c^2 .

Cohomology[edit]

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

H1(G, L×) = {1}.

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

H1(G,H) = {1}.

This is a generalization since L× = GL1(L).

Another generalization is H^1_{\acute{e}t}(X,\mathbf{G}_m) = H^1(X,\mathcal{O}_X^\times) = \mathrm{Pic}(X) for X a scheme, and another one to Milnor K-theory plays a role in Voevodsky's proof of the Milnor conjecture.

References[edit]