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2358. Barry Mazur, Karl Rubin and Alexandra Shlapentokh
Defining ℤ using unit groups
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Submission date: 4 March 2023

Abstract:

We consider first-order definability and decidability questions over rings of integers of algebraic extensions of ℚ, paying attention to the uniformity of definitions. The uniformity follows from the simplicity of our first-order definition of ℤ. Namely, we prove that for a large collection of algebraic extensions K/ℚ,

{ x ∈ O_K : ∀ ε ∈ O_K^× ∃ δ ∈ O_K^× such that δ-1 ≡(ε-1)x mod(ε-1)^2 } = ℤ
where O_K denotes the ring of integers of K.

Mathematics Subject Classification: 11U05

Keywords and phrases:

Full text arXiv 2303.02521: pdf, ps.