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SUMMARY:Russell Miller (City University of New York)
DTSTART:20240118T190000Z
DTEND:20240118T200000Z
DTSTAMP:20260423T021157Z
UID:OLS/143
DESCRIPTION:Title: Co
mputability and absolute Galois groups\nby Russell Miller (City Univer
sity of New York) as part of Online logic seminar\n\n\nAbstract\nThe ab
solute Galois group $\\operatorname{Gal}(F)$\nof a field $F$ is the Ga
lois group of its algebraic closure $\\overline{F}$\nrelative to $F$\, con
taining precisely those automorphisms of $\\overline{F}$\nthat fix $F$ its
elf pointwise. Even for a field as simple as the rational\nnumbers $\\mat
hbb{Q}$\, $\\operatorname{Gal}(\\mathbb Q)$ is a complicated\nobject. Ind
eed (perhaps counterintuitively)\, $\\operatorname{Gal}(\\mathbb Q)$\nis a
mong the thorniest of all absolute Galois groups normally studied.\n\nWhen
$F$ is countable\, $\\operatorname{Gal}(F)$ usually has the cardinality\n
of the continuum. However\, it can be presented as the set of all paths\n
through an $F$-computable finite-branching tree\, built by a procedure\nun
iform in $F$. We will first consider the basic properties of this tree\,\
nwhich depend in some part on $F$. Then we will address questions\nabout
the subgroup consisting of the computable paths through\nthis tree\, along
with other subgroups\nsimilarly defined by Turing ideals. One naturally
asks to what\nextent these are elementary subgroups of $\\operatorname{Gal
}(F)$\n(or at least elementarily equivalent to $\\operatorname{Gal}(F)$).\
nThis question is connected to the computability of Skolem functions\nfor
$\\operatorname{Gal}(F)$\, and also to the arithmetic complexity of\ndefin
able subsets of $\\operatorname{Gal}(F)$.\n\nSome of the results that will
appear represent joint work with\nDebanjana Kundu.\n
LOCATION:https://researchseminars.org/talk/OLS/143/
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