# Hilbert's Theorem 90

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 a cyclic extension of fields with Galois group G = Gal(L/K) generated by an element ${\displaystyle {\textbf {s}}}$, and if ${\displaystyle a}$ is an element of L of relative norm 1, then there exists ${\displaystyle b}$ in L such that

${\displaystyle a={\textbf {s}}(b)/b}$.

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:

${\displaystyle H^{1}(G,L^{\times })=\{1\}}$.

## ExamplesEdit

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

${\displaystyle s:\,\,c+di\mapsto c-di\ .}$

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

${\displaystyle y={\frac {c+di}{c-di}}={\frac {c^{2}-d^{2}}{c^{2}+d^{2}}}+{\frac {2cd}{c^{2}+d^{2}}}i}$

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

## CohomologyEdit

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

${\displaystyle H^{1}(G,L^{\times })=\{1\}.}$

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

${\displaystyle H^{1}(G,H)=\{1\}.}$

This is a generalization since ${\displaystyle L^{\times }=\mathbf {GL} _{1}(L)}$ .

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

## ProofEdit

### ElementaryEdit

Let ${\displaystyle L/K}$  be cyclic of degree ${\displaystyle n}$ , and ${\displaystyle \sigma }$  generate ${\displaystyle Gal(L/K)}$ .

Pick any ${\displaystyle a\in L}$  of norm ${\displaystyle N(a):=a\cdot \sigma (a)\cdot \sigma ^{2}(a)\cdots \sigma ^{n-1}(a)=1}$ .

• By clearing denominators, solving ${\displaystyle a=x/\sigma (x)}$  in ${\displaystyle L}$  is the same as showing that
${\displaystyle a\sigma (-)\ :L\longrightarrow L}$
has eigenvalue ${\displaystyle 1}$ .
• Extend this to a map of ${\displaystyle L}$ -vector spaces
${\displaystyle a\sigma (-)\ :L\otimes _{K}L\longrightarrow L\otimes _{K}L}$
by sending ${\displaystyle \ell \otimes \ell '\to \ell \otimes a\sigma (\ell ')}$ .

The primitive element theorem gives ${\displaystyle L=K(\alpha )}$  for some ${\displaystyle \alpha }$ . Since ${\displaystyle \alpha }$  has minimal polynomial ${\displaystyle f(t)=(t-\alpha )(t-\sigma (\alpha ))\cdots (t-\sigma ^{n-1}(\alpha ))\in K[t]}$ , we identify

${\displaystyle L\otimes _{K}L{\stackrel {\sim }{\longrightarrow }}L\otimes _{K}K[t]/f(t){\stackrel {\sim }{\longrightarrow }}L[t]/f(t){\stackrel {\sim }{\longrightarrow }}L^{n}}$

via ${\displaystyle \ell \otimes p(\alpha )\longrightarrow \ell (p(\alpha ),p(\sigma \alpha ),\cdots ,p(\sigma ^{n-1}\alpha ))}$ . Here we wrote the second factor as a ${\displaystyle K}$ -polynomial in ${\displaystyle \alpha }$ .

• Under this identification, our map
${\displaystyle a\sigma (-)\ :L^{n}\longrightarrow L^{n}}$
sends ${\displaystyle \ell \left(p(\alpha ),\cdots ,p(\sigma ^{n-1}\alpha )\right)\longrightarrow \ell (a\cdot p(\sigma \alpha ),\cdots ,\sigma ^{n-1}a\cdot p(\sigma ^{n}\alpha ))}$ . That is to say, this map sends ${\displaystyle (\ell _{1},...,\ell _{n})\longrightarrow (a\cdot \ell _{n},\sigma a\cdot \ell _{1},\cdots ,\sigma ^{n-1}a\cdot \ell _{n-1})}$ .
• ${\displaystyle (1,\sigma a,\sigma a\ \sigma ^{2}a,\cdots ,\sigma a\cdots \sigma ^{n-1}a)}$  is an eigenvector with eigenvalue ${\displaystyle 1}$  iff ${\displaystyle a}$  has norm ${\displaystyle 1}$ .