In
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
:
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
:
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
:
where ''h''
''i'' is the class number of ''K''
''i''. If one assumes that all the degrees