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Is there a prime number $p$, such that $\mathbb{Q}_p$ is isomorphic to a subfield of $\mathbb{C}$?

Eric Wofsey
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Mathemagician
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  • No. $\mathbb{C}$ is algebraically closed, whereas $\mathbb{Q}_p$ is not. You could ask the same question with respect to $\mathbb{C}$ and the algebraic closure of $\mathbb{Q}_p$, but the answer remains no, though for slightly more complicated reasons. – Xander Henderson Jul 16 '18 at 03:57
  • @Xander: I'm assuming that you mean "homeomorphic" when you read "homomorphic"? – Asaf Karagila Jul 16 '18 at 04:19
  • The idea was to ask if Q_p is isomorphic to some subfield of C, sorry if i expressed myself impropertly – Mathemagician Jul 16 '18 at 04:31
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    That usage of "homomorphic" is not correct. I have edited to make the meaning clear. – Eric Wofsey Jul 16 '18 at 04:47
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    https://math.stackexchange.com/questions/338148/is-there-an-explicit-embedding-from-the-various-fields-of-p-adic-numbers-mathb might be worth a look. – Gerry Myerson Jul 16 '18 at 07:24
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    I don't see an answer to this question at the "algebraically closed" question. Is this question really a duplicate of that one? – Gerry Myerson Jul 17 '18 at 05:38
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    if i understood correctly the statement of the question you gave me link to (and the top answer), the answer is yes for every p, but can't be explicitly constructed. – Mathemagician Jul 20 '18 at 19:56

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