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 ${\textbf {s}}$ , and if $a$ is an element of L of relative norm 1, then there exists $b$ in L such that

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:

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

.

Examples

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

s:\,\,c+di\mapsto c-di\ .

An element $x=a+bi$ in L has norm $xx^{s}=a^2 +b^2$ . An element of norm one corresponds to a rational solution of the equation $a^2+b^2=1$ 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 c, d) as

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 $\, (x,y)=(a/c,b/c)$ on the unit circle $x^2+y^2=1$ correspond to Pythagorean triples, i.e. triples $\,(a,b,c)$ of integers satisfying $\, a^2+b^2=c^2$ .

Cohomology

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

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

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

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

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

Proof

Elementary

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

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

By clearing denominators, solving $a=x/\sigma (x)$ in $L$ is the same as showing that

a\sigma (-)\ :L\longrightarrow L

has eigenvalue

1

.

Extend this to a map of $L$ -vector spaces

a\sigma (-)\ :L\otimes _{K}L\longrightarrow L\otimes _{K}L

by sending

\ell \otimes \ell '\to \ell \otimes a\sigma (\ell ')

.

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

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

Under this identification, our map

a\sigma (-)\ :L^{n}\longrightarrow L^{n}

sends

\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

(\ell _{1},...,\ell _{n})\longrightarrow (a\cdot \ell _{n},\sigma a\cdot \ell _{1},\cdots ,\sigma ^{n-1}a\cdot \ell _{n-1})

.

$(1,\sigma a,\sigma a\ \sigma ^{2}a,\cdots ,\sigma a\cdots \sigma ^{n-1}a)$ is an eigenvector with eigenvalue $1$ iff $a$ has norm $1$ .

References

Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456
Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62779-1, MR 1646901
Kummer, Ernst Eduard (1855), "Über eine besondere Art, aus complexen Einheiten gebildeter Ausdrücke.", Journal für die reine und angewandte Mathematik (in German), 50: 212–232, doi:10.1515/crll.1855.50.212, ISSN 0075-4102
Kummer, Ernst Eduard (1861), "Zwei neue Beweise der allgemeinen Reciprocitätsgesetze unter den Resten und Nichtresten der Potenzen, deren Grad eine Primzahl ist", Abdruck aus den Abhandlungen der Kgl. Akademie der Wissenschaften zu Berlin (in German), Reprinted in volume 1 of his collected works, pages 699–839
Chapter II of J.S. Milne, Class Field Theory, available at his website .
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, MR 1737196, Zbl 0948.11001
Noether, Emmy (1933), "Der Hauptgeschlechtssatz für relativ-galoissche Zahlkörper.", Mathematische Annalen (in German), 108 (1): 411–419, doi:10.1007/BF01452845, ISSN 0025-5831, Zbl 0007.29501
Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042

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