Timeline for Is a field that never embeds twice in another field necessarily a prime field?
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| Feb 28, 2018 at 10:43 | comment | added | YCor | A natural extension of the definition is to fix a field extension $k_0\to k$ and to say that $k$ is "unrepeatable" over $k_0$ if for every extension $k_0\to L$ there is at most one $k_0$-embedding $k\to L$. Then when $k_0$ is perfect, $k$ unrepeatable forces $k=k_0$. More generally this holds iff $k$ is a purely inseparable extension of $k_0$ (in the sense that $k$ belongs to the smallest perfect extension of $k_0$). I think this generality better emphasizes what happens (in the answer to the original question, prime fields being perfect, the role of perfectness doesn't show up explicitly). | |
| Feb 28, 2018 at 6:38 | vote | accept | Omar Antolín-Camarena | ||
| Feb 28, 2018 at 5:42 | answer | added | Sándor Kovács | timeline score: 15 | |
| Feb 28, 2018 at 5:21 | comment | added | Omar Antolín-Camarena | I definitely should have mentioned that, @GregMartin. (Minor correction: Frobenius endomorphism: it isn't always surjective. It's existence and non-identity-ness still shows unrepeatable positive characteristic fields are prime as you said.) | |
| Feb 28, 2018 at 5:03 | comment | added | Greg Martin | Non-prime finite fields are not unrepeatable, since the Frobenius $x\mapsto x^p$ gives a nontrivial automorphism of every $\Bbb F_{p^k}$ with $k\ge2$. Indeed, I believe this shows that no non-prime field of positive characteristic is unrepeatable. | |
| Feb 28, 2018 at 4:59 | history | edited | Greg Martin | CC BY-SA 3.0 |
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| Feb 28, 2018 at 4:55 | history | asked | Omar Antolín-Camarena | CC BY-SA 3.0 |