- Brauer-Siegel theorem
In
mathematics , the Brauer-Siegel theorem, named afterRichard Brauer andCarl Ludwig Siegel , is an asymptotic result on the behaviour ofalgebraic number field s, obtained byRichard Brauer andCarl Ludwig Siegel . It attempts to generalise the results known on theclass number s ofimaginary quadratic field s, to a more general sequence of number fields:"K"1, "K"2, ... .
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: ["K""i":Q] /log |"D""i"| → 0 as "i" → ∞.
Assuming that, and the algebraic hypothesis that "K""i" is a
Galois extension of Q, the conclusion is that:log ("h""i""R""i")/log √|"D""i"|) → 1 as "i" → ∞
where "h""i" is the class number of "K""i".
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", Amer. J. Math. 69 (1947), 243-250.
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