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Sep 22, 2020 at 17:17 vote accept Marc
Sep 22, 2020 at 16:05 comment added Marc Thanks! You are right, the question is subtly but seriously problematic. I actually mentioned in the question already that the action of $F_p$ is well-determined only up conjugation, but somehow I did not realize that this makes the question of whether $F_p(\Lambda_\ell) \subset \Lambda_\ell$ completely meaningless in most cases.
Sep 22, 2020 at 15:39 history answered Will Sawin CC BY-SA 4.0