Timeline for Definability of the ring of integer in algebraic extensions of $\mathbb Q$
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2021 at 18:11 | answer | added | Arno Fehm | timeline score: 4 | |
| Jun 4, 2019 at 2:13 | answer | added | Thomas Scanlon | timeline score: 15 | |
| Jun 2, 2019 at 8:34 | comment | added | YCor | @JoelDavidHamkins because when there's such an immediate answer, I expect from the OP a reaction such as "oh, I meant a finite algebraic extension". | |
| Jun 2, 2019 at 8:27 | comment | added | Joel David Hamkins | @YCor Why not post your comment as an answer? You have answered question 1, the point being that Tarski's elimination of quantifiers argument applies in any real-closed field, including the real algebraic numbers. So that is an algebraic extension in which $\mathbb{Z}$ is not definable, because the theory is decidable. | |
| Jun 1, 2019 at 15:24 | comment | added | YCor | Without further assumption on the algebraic extension, $\mathbf{Z}$ is not definable (even with constants) in the real-closed field $\mathbf{R}\cap\bar{\mathbf{Q}}$. | |
| Jun 1, 2019 at 14:33 | history | edited | YCor | CC BY-SA 4.0 |
clarified question, added tag
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| Jun 1, 2019 at 14:24 | comment | added | George Peterzil | @YCor Yes, that is what I mean | |
| Jun 1, 2019 at 14:21 | comment | added | YCor | Also "undecidable", does it mean here that the there's no Turing machine whose input is a free formula in the language of fields and whose output is yes/no according to whether it's true in $\mathbf{Q}$? | |
| Jun 1, 2019 at 14:15 | comment | added | YCor | The 1st § sounds a bit vague. Do you mean Robinson proved that there is a formula $F(x)$ in the language of fields such that for $x\in\mathbf{Q}$ we have $F(x)$ $\Leftrightarrow$ $x\in\mathbf{Z}$? In the question 1, by "the integers" do you mean $\mathbf{Z}$ or the ring of integers in $K$? | |
| Jun 1, 2019 at 14:10 | review | First posts | |||
| Jun 1, 2019 at 15:12 | |||||
| Jun 1, 2019 at 14:09 | history | asked | George Peterzil | CC BY-SA 4.0 |