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SUMMARY:Robert Lemke Oliver (Tufts University)
DTSTART:20241029T190000Z
DTEND:20241029T200000Z
DTSTAMP:20260419T151536Z
UID:BC-MIT/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/BC-MIT/13/">
 Enumerating Galois extensions of number fields</a>\nby Robert Lemke Oliver
  (Tufts University) as part of BC-MIT number theory seminar\n\nLecture hel
 d in Room 2-449 at MIT.\n\nAbstract\nLet $k$ be a number field. We provide
  an asymptotic formula for the number of Galois extensions of $k$ with abs
 olute discriminant bounded by some $X \\geq 1$ as $X \\to \\infty$. We als
 o provide an asymptotic formula for the closely related count of extension
 s $K/k$ whose normal closure has discriminant bounded by $X$. The key behi
 nd these results is a new upper bound on the number of Galois extensions o
 f $k$ with a given Galois group $G$ and discriminant bounded by $X$\; we s
 how the number of such extensions is $O_{[k:Q]\,G}(X^{4/\\sqrt{|G|}})$. Th
 is improves over the previous best bound $O_{k\,G\,\\epsilon}(X^{3/8+\\eps
 ilon})$ due to Ellenberg and Venkatesh. In particular\, ours is the first 
 bound for general $G$ with an exponent that decays as $|G| \\to \\infty$.\
 n
LOCATION:https://researchseminars.org/talk/BC-MIT/13/
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