Timeline for A ring map from algebraic integers to algebraic closure of $\mathbb F_p$
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| May 7, 2023 at 15:21 | vote | accept | UVIR | ||
| May 7, 2023 at 15:12 | history | edited | UVIR | CC BY-SA 4.0 |
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| May 7, 2023 at 3:32 | answer | added | Wojowu | timeline score: 9 | |
| May 7, 2023 at 0:18 | comment | added | Gerry Myerson | "monomial polynomials"? Do you mean, monic polynomials, UVIR? | |
| May 6, 2023 at 22:00 | answer | added | R. van Dobben de Bruyn | timeline score: 6 | |
| May 6, 2023 at 21:23 | history | edited | LSpice | CC BY-SA 4.0 |
Consistent notation
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| May 6, 2023 at 21:21 | comment | added | LSpice |
@UVIR, re, use Zorn to let $A$ be a maximal integral extension of $\mathbb Z$ such that there is a ring map $A \to \overline{\mathbb F_p}$, then show that, if $A$ is not all of $\overline{\mathbb Z}$, we can extend any given ring map $A \to \overline{\mathbb F_p}$ to a larger ring. \\ TeX note: \mathbb takes a following argument (unlike, e.g., \bf), so, e.g., {\mathbb F} is the same as {\mathbb{F}}, and you might as well drop the outer braces for \mathbb{F} (or \mathbb F).
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| May 6, 2023 at 20:32 | history | edited | UVIR | CC BY-SA 4.0 |
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| May 6, 2023 at 19:37 | review | Close votes | |||
| May 12, 2023 at 3:11 | |||||
| May 6, 2023 at 19:24 | comment | added | UVIR | @R. van Dobben de Bruyn Can you clarify a little bit more, maybe write an answer, or give me a reference? Thanks! | |
| May 6, 2023 at 19:14 | comment | added | R. van Dobben de Bruyn | Yes, $\operatorname{Hom}_{\text{Ring}}(\overline{\mathbf Z},\overline{\mathbf F}_p) \neq \varnothing$. In fact this set has a natural continuous action of $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$ by precomposition and by $\operatorname{Gal}(\overline{\mathbf F}_p/\mathbf F_p)$ by postcomposition. I think that the $\operatorname{Gal}(\overline{\mathbf Q}/\mathbf Q)$-action is transitive, and the stabiliser of a homomorphism is the inertia group of its kernel. | |
| May 6, 2023 at 18:59 | comment | added | UVIR | @R. van Dobben de Bruyn Then is there such a map, whether canonical or not? | |
| May 6, 2023 at 18:56 | comment | added | R. van Dobben de Bruyn | Every prime ideal $\mathfrak p \subseteq \overline{\mathbf Z}$ dividing $p\overline{\mathbf Z}$ has $\overline{\mathbf Z}/\mathfrak p \cong \overline{\mathbf F}_p$, but there is no preferred choice (neither of $\mathfrak p$ nor of this isomorphism). In fact, even saying the algebraic closure of $\mathbf F_p$ (or the set of algebraic integers) doesn't make sense; it is only defined up to non-canonical isomorphism. | |
| May 6, 2023 at 18:53 | history | asked | UVIR | CC BY-SA 4.0 |