Timeline for Algebraicity of the completion of a field? Finiteness?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2010 at 0:18 | comment | added | Pete L. Clark | @Dave: Yes, that's right. | |
| Jan 29, 2010 at 23:48 | comment | added | D. Savitt | Ah, I see, it's way easier than what I wrote below, isn't it? If an automorphism of C doesn't fix the reals, then the image of R contains a non-real z (R has no nontrivial automorphisms, so if an automorphism doesn't fix the reals then it doesn't preserve them), and then the Q-vector-subspace of the image generated by 1 and z is dense, so the whole image is dense. | |
| Jan 29, 2010 at 22:02 | comment | added | Pete L. Clark | Thanks again, Scott. Your new example sounds reasonable; I'll have to think about it. Now what about $[\hat{K}:K] \geq 4$? (Just kidding.) | |
| Jan 29, 2010 at 21:57 | history | edited | S. Carnahan♦ | CC BY-SA 2.5 |
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| Jan 29, 2010 at 21:24 | vote | accept | Pete L. Clark | ||
| Jan 29, 2010 at 21:24 | comment | added | Pete L. Clark | My (apparently MO-shy) student came up with the example of a subextension L of R/Q such that L/Q is purely transcendental and R/L is (certainly nontrivial) algebraic. As I pointed out in today's problem session, the Artin-Schreier theorem guarantees that R/L is infinite, so this does not answer the second question. In fact, your very nice answer to the second question prompts a third question: can we have $2 < [\hat{K}:K] < \infty$? | |
| Jan 29, 2010 at 14:53 | history | edited | S. Carnahan♦ | CC BY-SA 2.5 |
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| Jan 29, 2010 at 14:48 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |