Hilbert's Theorem 90

# Hilbert's Theorem 90

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In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, Hilbert's Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

s (or to one of its generalizations) that leads to Kummer theory
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...

. In its most basic form, it tells us 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
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

's famous Zahlbericht
Zahlbericht
In mathematics, the Zahlbericht was a report on algebraic number theory by .-History: and and the English introduction to give detailed discussions of the history and influence of 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 Galois group G =Gal(L/K), then the first cohomology group is trivial:
H1(G, L×) = {1}

## Examples

Let L/K be the quadratic extension . The Galois group is cyclic of order 2, its generator s is acting via conjugation:
An element in L has norm . 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
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

. Hilbert's Theorem 90 then states that every 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 triple
Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...

s, i.e. triples of integers satisfying .

## Cohomology

The theorem can be stated in terms of group cohomology
Group cohomology
In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...

: if L× is the multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...

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
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

or special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....

over L, then
H1(G,H) = {1}.

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

Another generalization is for X a scheme, and another one to Milnor K-theory
Milnor K-theory
In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by .The calculation of K2 of a field k led Milnor to the following ad hoc definition of "higher" K-groups by...

plays a role in Voevodsky's