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Jan 18, 2022 at 20:28 comment added Will Sawin @kiran For the first case, yes, by the same proof. For the second case, you have to use a more complicated proof (for example looking at the set of all finite extensions of $k$) but the answer is still yes.
Jan 18, 2022 at 19:58 comment added kiran Thanks! Does a similar result hold for F_q, q=p^k? Or even more ambitiously, for k a random perfect field of char p?
Jan 14, 2022 at 2:25 vote accept kiran
Jan 13, 2022 at 23:34 history answered Will Sawin CC BY-SA 4.0