Timeline for Reference for countable and uncountable algebraic closures of $\mathbb{Q}$ in ZF
Current License: CC BY-SA 4.0
2 events
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| Jan 2, 2023 at 11:27 | comment | added | Asaf Karagila♦ | I mean, (2) is a very easy consequence of the ZFC result that all algebraic closures of a fixed field are isomorphic. If $F_1,F_2$ are two well-orderable algebraic closures of a well-orderable field $K$, there are sets of ordinals $A_1,A_2$ and $A_K$ which code these structures, then in $L[A_1,A_2,A_K]$ the field $K$ exists, and both $F_i$ exist and are algebraically closed, so they are isomorphic. Of course, if $K$ is $\Bbb Q$ or $\Bbb F_p$ we can ignore $A_K$ altogether, and if $F_i$ is countable, we can just take $A_i$ to be some subset of $\omega$. | |
| Jul 13, 2022 at 14:36 | history | asked | Gro-Tsen | CC BY-SA 4.0 |