Hilbert's Theorem 90

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In number theory, Hilbert's Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of number fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it 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 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 Hilbert's famous Zahlbericht of 1897, although it is originally due to Kummer. Often a more general theorem due to Emmy Noether 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}

[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 xxs = a2 + b2. 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 x2 + y2 = 1 correspond to Pythagorean triples, i.e triples \,(a,b,c) of integers satisfying \, a^2+b^2=c^2 .

[edit] Cohomology

The theorem can be stated in terms of group cohomology: if L× is the multiplicative group of L, then

H1(G, L×) = {1}

provided that G is cyclic.


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).


[edit] References

  • Chapter II of J.S. Milne, Class Field Theory, available at his website [1].
  • J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, Grundlehren der mathematischen Wissenschaften. vol. 323, Springer-Verlag, 2000. ISBN 3-540-66671-0