Hilbert's Theorem 90

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In number theory, Hilbert's Theorem 90 (or Satz 90) tells us that if L/K is a cyclic extension of number fields with Galois group G =Gal(L/K) generated by an element s and if α is an element of L of relative norm 1, then there exists β in L such that

α = β/β^s.

The theorem can be stated in terms of group cohomology: if L^x is the multiplicative group of L, then the Tate cohomology group is trivial:

H⁻¹(G, L^x) = {1}.

(provided that G is cyclic).

The theorem takes its name from the fact that it is the 90th theorem in Hilbert's famous Zahlbericht of 1897, although it is originally due to Kummer. Often a more general theorem due to Emmy Nöther is given the name, stating that if L/K is a finite Galois extension of fields, then the first cohomology group is trivial;

H¹(G, L^x) = {1}

This generalizes the theorem for cyclic extensions, because if G is a cyclic group then the groups H⁻¹(G, M) and H¹(G, M) are isomorphic for any G -module M (and more generally the Tate cohomology groups are periodic of period 2).

[edit] Examples

Let L/Q be the quadratic extension $\mathbb{Q}(\sqrt{-1})$ . The Galois group is cyclic of order 2, its generator s is acting via conjugation:

$s:\, \, a+b\sqrt{-1}\mapsto a-b\sqrt{-1}\ .$

An element $x=a+b\sqrt{-1}$ in L has norm $x x s = a 2 + b 2$ . An element of norm one corresponds to a rational solution of the equation a² +b²=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 rational c,d) as

$y={{c+d\sqrt{-1}}\over{c-d\sqrt{-1}}}={{c^2-d^2}\over{c^2+d^2}}+{2dc\over{c^2+d^2}}\sqrt{-1}$

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 $\, a^2+b^2=c^2$ .