Is there an isomorphism from $\overline{\mathbb{Q}}_p$ to $\mathbb{C}$ as fields?

Question

Is there an isomorphism from $\overline{\mathbb{Q}}_p$ to $\mathbb{C}$ as fields?

It seems that since they are both algebraically closed and characteristic 0, there is an isomorphism. However, I am not able to actually construct the isomorphism, nor is it clear that such an isomorphism must exist.

See also this MO-question. – Dietrich Burde May 05 '17 at 19:05 — Dietrich Burde, May 05 '17 at 19:05
Thank you very much @DietrichBurde – Miko Himmel May 05 '17 at 20:47 — Miko Himmel, May 05 '17 at 20:47

score 6 · Answer 1 · answered May 05 '17 at 17:47

6

If you believe the axiom of choice, yes there is. They are algebraically closed fields of the same characteristic, and the same uncountable cardinality. Therefore they both have the same transcendence degree over $\Bbb Q$. Zorn's lemma can be used to prove an isomorphism between them, just not one that you can actually write down.

answered May 05 '17 at 17:47

Angina Seng

158,341

There isn't actually a theorem that you can't write one down. In fact, it's possible that every set can be written down. – May 05 '17 at 21:47

Is there an isomorphism from $\overline{\mathbb{Q}}_p$ to $\mathbb{C}$ as fields?

1 Answers1