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A topological approach to undefinability in algebraic extensions of
Q
\mathbb{Q}
Q
Authors
Kirsten Eisentraeger
Russell Miller
Caleb Springer
Linda Westrick
Publication date
27 October 2023
Publisher
View
on
arXiv
Abstract
For any subset
Z
β
Q
Z \subseteq \mathbb{Q}
Z
β
Q
, consider the set
S
Z
S_Z
S
Z
β
of subfields
L
β
Q
βΎ
L\subseteq \overline{\mathbb{Q}}
L
β
Q
β
which contain a co-infinite subset
C
β
L
C \subseteq L
C
β
L
that is universally definable in
L
L
L
such that
C
β©
Q
=
Z
C \cap \mathbb{Q}=Z
C
β©
Q
=
Z
. Placing a natural topology on the set
Sub
(
Q
βΎ
)
\text{Sub}(\overline{\mathbb{Q}})
Sub
(
Q
β
)
of subfields of
Q
βΎ
\overline{\mathbb{Q}}
Q
β
, we show that if
Z
Z
Z
is not thin in
Q
\mathbb{Q}
Q
, then
S
Z
S_Z
S
Z
β
is meager in
Sub
(
Q
βΎ
)
\text{Sub}(\overline{\mathbb{Q}})
Sub
(
Q
β
)
. Here, thin and meager both mean "small", in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields
L
L
L
have the property that the ring of algebraic integers
O
L
\mathcal{O}_L
O
L
β
is universally definable in
L
L
L
. The main tools are Hilbert's Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every
β
\exists
β
-definable subset of an algebraic extension of
Q
\mathbb Q
Q
is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.Comment: 24 pages. Introduction has been rewritte
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Last time updated on 30/10/2020