94 research outputs found
Reverse Mathematics and Algebraic Field Extensions
This paper analyzes theorems about algebraic field extensions using the
techniques of reverse mathematics. In section 2, we show that
is equivalent to the ability to extend -automorphisms of field extensions to
automorphisms of , the algebraic closure of . Section 3 explores
finitary conditions for embeddability. Normal and Galois extensions are
discussed in section 4, and the Galois correspondence theorems for infinite
field extensions are treated in section 5.Comment: 25 page
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Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
Infinite time decidable equivalence relation theory
We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic. We also introduce an infinite time generalization of the countable
Borel equivalence relations, a key subclass of the Borel equivalence relations,
and again show that several key properties carry over to the larger class.
Lastly, we collect together several results from the literature regarding Borel
reducibility which apply also to absolutely Delta_1^2 reductions, and hence to
the infinite time computable reductions.Comment: 30 pages, 3 figure
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