Hilbert's Theorem 90

The mathematical theorem , which David Hilbert lists under number 90 in his theory of algebraic number fields and which has been called this since then, makes a statement about the structure of certain field extensions . It was proven en passant by Kummer as early as 1855 .

Original version

Let it be a cyclical Galois expansion and a producer of the associated Galois group . Then each with norm is of the form ${\displaystyle L/K}$ ${\displaystyle \sigma }$ ${\displaystyle y\in L^{\times }}$ ${\displaystyle N_{L/K}(y)=1}$

${\displaystyle y={\frac {\sigma x}{x}}}$

with a suitable . ${\displaystyle x\in L^{\times }}$

Galois cohomological version

${\displaystyle E}$is a body , a Galois body extension and . Then follows for the Galois cohomology : ${\displaystyle E/F}$ ${\displaystyle G={\text{Gal}}(E/F)}$

${\displaystyle H^{1}(G,E^{\times })=0}$

Algebraic-geometric frame

It is a scheme . Then ${\displaystyle X}$

${\displaystyle \mathrm {H} _{\mathrm {{\acute {e}}t} }^{1}(X,\mathbb {G} _{\mathrm {m} })=\mathrm {Pic} \,X.}$

In other words: every étale-locally trivial line bundle is already a Zariski line bundle.

Hilbert 90 for motivic cohomology

The original version generalizes in the motivic cohomology to the exactness of

${\displaystyle \mathrm {H} ^{1}(Y,\mathbb {G} _{\mathrm {m} })\to ^{1-\sigma }H^{1}(Y,\mathbb {G} _{\mathrm {m} })\to ^{N_{X/Y}}H^{1}(X,\mathbb {G} _{\mathrm {m} })}$

for cyclic Galois overlays with generator . For the spectrum of a body you get back the original version. ${\displaystyle Y/X}$ ${\displaystyle \sigma }$

literature

• David Hilbert : The theory of algebraic number fields. (PDF; 90 MB). In: Payment report. Annual report of the German Mathematicians Association, Vol. 4, pp. 175–546, 1897, see p. 272.

Web links

Wikisource: Hilbert's sentence 90. In: David Hilbert, Gesammelte Abhandlungen, Volume 1  - Sources and full texts

Individual evidence

1. ^ Franz Lemmermeyer (2018): 120 years of Hilbert's number report. (PDF; 541 kB). Annual reports of the German Mathematicians Association, 120 (1), 41–79, see p. 10.