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Brauer-Siegel Theorem

The Brauer-Siegel theorem relates the class number h_K, regulator R_K, and absolute number field discriminant d_K of a number field K. For a family of Galois extensions of Q with d_K->infty and [K:Q]/lnd_K->0, it gives

ln(h_KR_K)∼1/2lnd_K

(Banks 2026).

Banks (2026) reports the effective lower bound

h_KR_K>=(c(k)sqrt(d_K))/((2pi)^([K:Q])lnd_K)

for every Galois extension field K/k with K!=Q, where the base number field k is fixed and c(k)>0 is effective. The constant is not asserted to be absolute over varying base fields. The argument also reports an effective Landau-Siegel zero gap for quadratic extensions of k. These new effectiveness claims require further independent verification. Banks (2026) credits Claude with an initial argument from the author's ideas and with proof checking.

See also

Class Number, Landau-Siegel Zero, Number Field Discriminant, Regulator

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References

Banks, W. D. "Landau-Siegel Zeros over a Base Field and Effective Brauer-Siegel." 22 Oct 2026. https://arxiv.org/abs/2610.31402.Lang, S. Algebraic Number Theory, 2nd ed. New York: Springer-Verlag, pp. 321-330, 1994. https://doi.org/10.1007/978-1-4612-0853-2.

Cite this as:

Weisstein, Eric W. "Brauer-Siegel Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Brauer-SiegelTheorem.html

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