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Concept
Hilbert's Theorem 90
Summary
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 an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element and if is an element of L of relative norm 1, that isthen there exists in L such thatThe theorem takes its name from the fact that it is the 90th theorem in David Hilbert's Zahlbericht , although it is originally due to .
Often a more general theorem due to is given the name, stating that if L/K is a finite Galois extension of fields with arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial:
Let be the quadratic extension . The Galois group is cyclic of order 2, its generator acting via conjugation:
An element in has norm . An element of norm one thus 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 a of norm one can be written as
where is as in the conclusion of the theorem, and c and d are both integers. This 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 .
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
Specifically, group cohomology is the cohomology of the complex whose i-cochains are arbitrary functions from i-tuples of group elements to the multiplicative coefficient group, , with differentials defined in dimensions by:
where denotes the image of the -module element under the action of the group element .
Note that in the first of these we have identified a 0-cochain , with its unique image value .
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