As we all know, the complex number field $\mathbb{C}$ be a finite Galois extension field of the real number field that contains all algebraic numbers.
I want to know the proof of the following proposition:
Any p-adic number field $\mathbb{Q}\_p$ has a finite Galois extension field $E$ such that $E$ contains $\overline{\mathbb{Q}}$.
In addition, what is the relationship between the complex number field and the algebraic closed of $\mathbb{Q}\_p$?
Is there an extension field of $\mathbb{Q}$, it can be embedded into $\mathbb{Q}\_p$ and can be embedded into the real number field? If the answer is yes, is there the largest such field?



$p$-adic number field $\mathbb{Q}_p $and algebraic numbers
I am trying to answer what it seems you ask.


About your last question: "Is there an extension field of $\mathbb{Q}$, such that it can be embedded into $\mathbb{Q}\_p$ and it can be embedded into the real number field? If the answer is yes, is there the largest such field?"


If you consider **algebraic** extensions, then for any prime number $p$ there is the so called maximal totally $p$-adic extension, formed by the algebraic numbers (in $\overline{\mathbb{Q}}$) whose irreducible polynomial splits in $\mathbb{Q}\_p$. It is an extension of infinite degree. It can be seen as the maximal algebraic extension of $\mathbb{Q}$ inside $\mathbb{Q}\_p$.


One can also take the intersection (inside $\overline{\mathbb{Q}}$) of the maximal totally $p$-adic extensions for some primes and also of the totally real algebraic numbers (which satisfy the analogous property to "totally $p$-adic" but for the real field), and if you intersect finitely many of them the result is still an infinite algebraic extension of $\mathbb{Q}$.


If you consider arbitrary subfields of the real numbers and of the $p$-adic numbers, as already said, you cannot even compare them since there is no "natural" field containing both ($p$-adic and reals).


About the proof of the proposition you want to know, I am sorry to say that there is no such proof as the proposition you stated is false. One has that for any $p$-adic field $\mathbb{Q}\_p$, there is no finite extension containing $\overline{\mathbb{Q}}$.


