Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diophantine Definability and Decidability in the Extensions of Degree 2
  of Totally Real Fields   We investigate Diophantine definability and decidability over some subrings
of algebraic numbers contained in quadratic extensions of totally real
algebraic extensions of $\mathbb Q$. Among other results we prove the
following. The big subring definability and undecidability results previously
shown by the author to hold over totally complex extensions of degree 2 of
totally real number fields, are shown to hold for {\it all} extensions of
degree 2 of totally real number fields. The definability and undecidability
results for integral closures of ``small'' and ``big'' subrings of number
fields in the infinite algebraic extensions of $\mathbb Q$, previously shown by
the author to hold for totally real fields, are extended to a large class of
extensions of degree 2 of totally real fields. This class includes infinite
cyclotomics and abelian extensions with finitely many ramified rational primes.

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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
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The categories of this paper are: solv-int