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SUMMARY:Brian Tyrell (University of Oxford)
DTSTART:20230208T160000Z
DTEND:20230208T170000Z
DTSTAMP:20260418T064738Z
UID:LNTS/94
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/94/">Fu
 rther afield and further\, a field: remarks on undecidability</a>\nby Bria
 n Tyrell (University of Oxford) as part of London number theory seminar\n\
 nLecture held in Rm. 706\, UCL Department of Mathematics (UCL Union Buildi
 ng).\n\nAbstract\nGiven a field $K$\, one can ask "what first-order senten
 ces are true in $K$"? E.g. for $K = \\mathbb{C}$\, "$\\exists x (x^2 = -1)
 $" is true\, but for $K = \\mathbb{Q}$ this is false. One major area of st
 udy at the intersection of logic and number theory is\, given a field $K$ 
 of number-theoretic interest\, whether there is an algorithmic process whi
 ch can decide the truth or falsity of a given first-order sentence in $K$.
  For $K = \\mathbb{C}$\, there exists such an algorithmic process\; for $K
  = \\mathbb{Q}$ there cannot (due to work of Gödel & Julia Robinson).\n\n
 I will pose a related question: whether the logical consequences of a give
 n sentence in a field may be decided algorithmically. Often the answer is 
 no\; so e.g. we cannot algorithmically detect general properties of fields
  $K$ with a Galois extension $L$ such that $\\mathrm{Gal}(L/K) \\cong S_5$
 \, or e.g. general properties of characteristic $p$ fields that admit poin
 ts on a given rationally parameterisable curve over $\\mathbb{F}_p$. I wil
 l focus on those fields whose behaviour is tightly controlled by their abs
 olute Galois group\, and prove some precise limitations.\n\nI will aim for
  this talk to be self-contained on the logic side of things!\n
LOCATION:https://researchseminars.org/talk/LNTS/94/
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