Hilbert's Theorem 90

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

:$\backslash sigma:\; c\; +\; di\backslash mapsto\; c\; -\; di.$

An element $a=x+yi$ in $\backslash Q(i)$ has norm $a\backslash sigma(a)=x^2+y^2$. An element of norm one thus corresponds to a rational solution of the equation $x^2+y^2=1$ or in other words, a point with rational coordinates on the unit circle. Hilbert's Theorem 90 then states that every such element a of norm one can be written as

:$a=\backslash frac\{c-di\}\{c+di\}=\backslash frac\{c^2-d^2\}\{c^2+d^2\}\; -\; \backslash frac\{2cd\}\{c^2+d^2\}\; i,$

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

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

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

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, $C^i(G,L^\backslash times)\; =\; \backslash \{\backslash phi:G^i\backslash to\; L^\backslash times\backslash \}$, with differentials $d^i\; :\; C^i\backslash to\; C^\{i+1\}$ defined in dimensions $i\; =\; 0,1$ by:

$(d^0(b))(\backslash sigma)\; =\; b\; /\; b^\backslash sigma,\; \backslash quad\; \backslash text\{\; and\; \}$

\quad (d^1(\phi))(\sigma,\tau)

\,=\,\phi(\sigma) \phi(\tau)^\sigma / \phi(\sigma\tau) ,

where $x^g$ denotes the image of the $G$-module element $x$ under the action of the group element $g\; \backslash in\; G$.

Note that in the first of these we have identified a 0-cochain $\backslash gamma\; =\; \backslash gamma\_b\; :\; G^0\; =\; id\_G\; \backslash to\; L^\backslash times$, with its unique image value $b\; \backslash in\; L^\backslash times$.

The triviality of the first cohomology group is then equivalent to the 1-cocycles $Z^1$ being equal to the 1-coboundaries $B^1$, viz.:

$\backslash begin\{array\}\{rcl\}$

Z^1 &=& \ker d^1 &=& \{\phi\in C^1\text{ satisfying }\,\,\forall \sigma , \tau \in G \, \colon\,\, \phi(\sigma\tau) = \phi(\sigma)\,\phi(\tau)^\sigma \} \\

\text{ is equal to }\\

B^1 &=& \text{im } d^0 &=& \{\phi\in C^1\ \, \colon \,\, \exists\, b\in L^\times \text{ such that } \phi(\sigma)=b / b^\sigma \ \ \forall \sigma \in G \}.

\end{array}

For cyclic $G\; =\backslash \{1,\backslash sigma,\backslash ldots,\backslash sigma^\{n-1\}\backslash \}$, a 1-cocycle is determined by $\backslash phi(\backslash sigma)=a\backslash in\; L^\backslash times$, with $\backslash phi(\backslash sigma^i)\; =\; a\backslash ,\backslash sigma(a)\backslash cdots\backslash sigma^\{i-1\}(a)$ and:

$1=\backslash phi(1)=\backslash phi(\backslash sigma^n)=a\backslash ,\backslash sigma(a)\backslash cdots\backslash sigma^\{n-1\}(a)=N(a).$

A further generalization is to cohomology with non-abelian coefficients: that if H is either the general or special linear group over L, including $\backslash operatorname\{GL\}\_1(L)=L^\backslash times$, then

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

:$H^1\_\{\backslash text\{et\}\}(X,\backslash mathbb\{G\}\_m)\; =\; H^1(X,\backslash mathcal\{O\}\_X^\backslash times)\; =\; \backslash operatorname\{Pic\}(X),$

where $\backslash operatorname\{Pic\}(X)$ is the group of isomorphism classes of locally free sheaves of $\backslash mathcal\{O\}\_X^\backslash times$-modules of rank 1 for the Zariski topology, and $\backslash mathbb\{G\}\_m$ is the sheaf defined by the affine line without the origin considered as a group under multiplication. {{Cite web|last=Milne|first=James S.|date=2013|title=Lectures on Etale Cohomology (v2.21)|url=https://www.jmilne.org/math/CourseNotes/LEC.pdf|page=80}}

There is yet another generalization to Milnor K-theory which plays a role in Voevodsky's proof of the Milnor conjecture.

Let $L/K$ be cyclic of degree $n,$ and $\backslash sigma$ generate $\backslash operatorname\{Gal\}(L/K)$. Pick any $a\backslash in\; L$ of norm

:$N(a):=a\; \backslash sigma(a)\; \backslash sigma^2(a)\backslash cdots\; \backslash sigma^\{n-1\}(a)=1.$

By clearing denominators, solving $a=x/\backslash sigma^\{-1\}(x)\; \backslash in\; L$ is the same as showing that $a\backslash sigma^\{-1\}(\backslash cdot)\; :\; L\; \backslash to\; L$ has $1$ as an eigenvalue. We extend this to a map of $L$-vector spaces via

:$\backslash begin\{cases\}\; 1\_L\backslash otimes\; a\backslash sigma^\{-1\}(\backslash cdot)\; :\; L\backslash otimes\_KL\; \backslash to\; L\backslash otimes\_K\; L\; \backslash \backslash \; \backslash ell\; \backslash otimes\backslash ell\text{'}\backslash mapsto\; \backslash ell\backslash otimes\; a\backslash sigma^\{-1\}(\backslash ell\text{'}).\backslash end\{cases\}$

The primitive element theorem gives $L=K(\backslash alpha)$ for some $\backslash alpha$. Since $\backslash alpha$ has minimal polynomial

:$f(t)=(t-\backslash alpha)(t-\backslash sigma(\backslash alpha))\backslash cdots\; \backslash left\; (t-\backslash sigma^\{n-1\}(\backslash alpha)\; \backslash right\; )\; \backslash in\; K[t],$

we can identify

:$L\backslash otimes\_KL\backslash stackrel\{\backslash sim\}\{\backslash to\}\; L\backslash otimes\_K\; K[t]\; /f(t)\; \backslash stackrel\{\backslash sim\}\{\backslash to\}\; L[t]/f(t)\; \backslash stackrel\{\backslash sim\}\{\backslash to\}\; L^\{n\}$

via

:$\backslash ell\backslash otimes\; p(\backslash alpha)\; \backslash mapsto\; \backslash ell\; \backslash left\; (p(\backslash alpha),\; p(\backslash sigma\; \backslash alpha\; ),\backslash ldots\; ,\; p(\backslash sigma^\{n-1\}\; \backslash alpha\; )\; \backslash right\; ).$

Here we wrote the second factor as a $K$-polynomial in $\backslash alpha$.

Under this identification, our map becomes

:$\backslash begin\{cases\}\; a\backslash sigma^\{-1\}(\backslash cdot)\; :\; L^n\backslash to\; L^n\; \backslash \backslash \; \backslash ell\; \backslash left(p(\backslash alpha),\backslash ldots,\; p(\backslash sigma^\{n-1\}\backslash alpha))\; \backslash mapsto\; \backslash ell(ap(\backslash sigma^\{n-1\}\backslash alpha),\; \backslash sigma\; a\; p(\backslash alpha),\; \backslash ldots,\; \backslash sigma^\{n-1\}\; a\; p(\backslash sigma^\{n-2\}\backslash alpha)\; \backslash right\; ).\; \backslash end\{cases\}$

That is to say under this map

:$(\backslash ell\_1,\; \backslash ldots\; ,\backslash ell\_n)\backslash mapsto\; (a\; \backslash ell\_n,\; \backslash sigma\; a\; \backslash ell\_1,\; \backslash ldots,\; \backslash sigma^\{n-1\}\; a\; \backslash ell\_\{n-1\}).$

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

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{{wikisource|de:Seite:David Hilbert Gesammelte Abhandlungen Bd 1.djvu/166|Hilbert's Theorem 90 in: David Hilbert, Gesammelte Abhandlungen, Erster Band}}

Category:Theorems in algebraic number theory