Brauer–Siegel Theorem
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Brauer–Siegel theorem, named after
Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...
and
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
, is an asymptotic result on the behaviour of
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
s, obtained by
Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...
and
Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, ...
. It attempts to generalise the results known on the class numbers of
imaginary quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...
s, to a more general sequence of number fields :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 In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...
. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if ''D''''i'' is the
discriminant In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the zero of a function, roots without computing them. More precisely, it is a polynomial function of the coef ...
of ''K''''i'', then : \frac \to 0\texti \to\infty. Assuming that, and the algebraic hypothesis that ''K''''i'' is a
Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...
of Q, the conclusion is that : \frac \to 1\texti \to\infty where ''h''''i'' is the class number of ''K''''i''. If one assumes that all the degrees _i : \mathbf Q/math> 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 Harold Mead Stark (born August 6, 1939) is an Americans, American mathematician, specializing in number theory. He is best known for his solution of the Carl Friedrich Gauss, Gauss class number 1 problem, in effect Stark–Heegner theorem, corre ...
from the early 1970s.


References

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Richard Brauer Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation t ...
, ''On the Zeta-Function of Algebraic Number Fields'', ''
American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
'' 69 (1947), 243–250. Analytic number theory Theorems in algebraic number theory {{numtheory-stub