Brauer–Siegel theorem

In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the results known on the class numbers of imaginary quadratic fields, to a more general sequence of number fields

${\displaystyle K_{1},K_{2},\ldots .\ }$

In all cases other than the rational field Q and imaginary quadratic fields, the regulator R_i of K_i must be taken into account, because K_i then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if D_i is the discriminant of K_i, then

${\displaystyle {\frac {[K_{i}:\mathbf {Q} ]}{\log |D_{i}|}}\to 0{\text{ as }}i\to \infty .}$

Assuming that, and the algebraic hypothesis that K_i is a Galois extension of Q, the conclusion is that

${\displaystyle {\frac {\log(h_{i}R_{i})}{\log {\sqrt {|D_{i}|}}}}\to 1{\text{ as }}i\to \infty }$

where h_i is the class number of K_i. If one assumes that all the degrees ${\displaystyle [K_{i}:\mathbf {Q} ]}$ are bounded above by a uniform constant N, then one may drop the assumption of normality - this is what is actually proved in Brauer's paper.

This result is ineffective, as indeed was the result on quadratic fields on which it built. Effective results in the same direction were initiated in work of Harold Stark from the early 1970s.

References

• Richard Brauer, On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69 (1947), 243–250.