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For each positive integer $n$, let $C_n$ for the minimum root discriminant for all number fields of degree $n$. Assuming the Generalized Riemann Hypothesis, $\limsup C_n \geq \Omega$ where $$ \Omega = 8\pi e^\gamma\approx 44.7632\ldots$$ and $\gamma$ is the Euler–Mascheroni constant. Lower bounds for the $C_n$ were deduced by analytic methods through the work of Odlyzko and others. In particular, Serre introduced the constant $\Omega$ which we refer to as the Serre Odlyzko bound,

Consequently, any number field whose root discriminant lies below $\Omega$ can be considered to have small discriminant.

Knowl status:
  • Review status: reviewed
  • Last edited by David Roberts on 2019-05-03 20:34:04
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