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| In [[mathematics]], the '''Brauer–Siegel theorem''', named after [[Richard Brauer]] and [[Carl Ludwig Siegel]], is an asymptotic result on the behaviour of [[algebraic number field]]s, obtained by [[Richard Brauer]] and [[Carl Ludwig Siegel]]. It attempts to generalise the results known on the [[class number (number theory)|class number]]s of [[imaginary quadratic field]]s, to a more general sequence of number fields
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| :<math>K_1, K_2, \ldots.\ </math>
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| In all cases other than the rational field '''Q''' and imaginary quadratic fields, the [[regulator of a number field|regulator]] ''R''<sub>''i''</sub> of ''K''<sub>''i''</sub> must be taken into account, because ''K''<sub>i</sub> then has units of infinite order by [[Dirichlet's unit theorem]]. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if ''D''<sub>''i''</sub> is the [[discriminant of an algebraic number field|discriminant]] of ''K''<sub>''i''</sub>, then
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| : <math> \frac{[K_i : Q]}{\log|D_i|} \to 0\text{ as }i \to\infty. </math>
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| Assuming that, and the algebraic hypothesis that ''K''<sub>''i''</sub> is a [[Galois extension]] of '''Q''', the conclusion is that
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| : <math> \frac{ \log(h_i R_i) }{ \log\sqrt{|D_i|} } \to 1\text{ as }i \to\infty </math> | |
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| where ''h''<sub>''i''</sub> is the class number of ''K''<sub>''i''</sub>.
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| This result is [[effective results in number theory|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.
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| ==References==
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| * [[Richard Brauer]], ''On the Zeta-Function of Algebraic Number Fields'', ''[[American Journal of Mathematics]]'' 69 (1947), 243–250.
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| {{DEFAULTSORT:Brauer-Siegel theorem}}
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| [[Category:Analytic number theory]]
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| [[Category:Theorems in algebraic number theory]]
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