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SUMMARY:Bhawesh Mishra (University of Memphis)
DTSTART:20231205T210000Z
DTEND:20231205T220000Z
DTSTAMP:20260423T024838Z
UID:UAANTS/81
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UAANTS/81/">
 A Generalization of the Grunwald-Wang Theorem for $n^{th}$ Powers</a>\nby 
 Bhawesh Mishra (University of Memphis) as part of University of Arizona Al
 gebra and Number Theory Seminar\n\n\nAbstract\nThe Grunwald-Wang theorem f
 or $n^{th}$ powers states that a rational number $a$ is an $n^{th}$ power 
 in $\\mathbb{Q}_{p}$ for almost every prime $p$ if and only if either $a$ 
 is a perfect $n^{th}$ power in rationals or $8 \\mid n$ and $a = 2^{\\frac
 {n}{2}} \\cdot b^{n}$ for some rational $b$. We will discuss an appropriat
 e generalization of this theorem from a single rational number to a subset
  $A$ of rational numbers. Let $n$ be an odd number and $q$ be the smallest
  prime dividing $n$. A finite subset $A$ of rationals with cardinality $\\
 leq q$ contains a $n^{th}$ power in $\\mathbb{Q}_{p}$ for almost every pri
 me $p$ if and only if $A$ contains a perfect $n^{th}$ power. For even $n$\
 , the result is analogous - up to a short list of exceptions\, as evident 
 in the Grunwald-Wang theorem. \n\nIf time permits\, we will also show that
  this generalization is optimal\, i.e.\, for every $n \\geq 2$\, there are
  infinitely many subsets $A$ of rationals of cardinality $q+1$ that contai
 n a $n^{th}$ power in $\\mathbb{Q}_{p}$ for almost every prime $p$ but nei
 ther contain a perfect $n^{th}$ power nor fall under the finite list of ex
 ceptions.\n
LOCATION:https://researchseminars.org/talk/UAANTS/81/
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