What does mean by "algebraic closure of $\mathbb Q$ in $\mathbb Q_p$"?

Question

I want to understand the precise meaning of "algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_p$".

• What does mean by "algebraic closure of $\mathbb Q$ in $\mathbb Q_p$" ? Isn't it just the set $\bar{\mathbb Q} \cap \mathbb Q_p$ ?

This set may be think of as "set of algebraic numbers" or even "set of $p$-adic algebraic numbers". It is closed field but not complete with respect $p$-adic topology.

Thanks

Answer 1

You can't take the intersection of $\bar{\mathbb Q}$ and $\mathbb Q_p$ because these aren't subsets of some common ambient set.

By definition if $L/K$ is a field extension (not necessarily algebraic) then you define the "algebraic closure of $L$ in $K$" as the set

$$\{\alpha\in L\mid\text{$\alpha$ is algebraic over $K$}\}$$

which is always a subfield of $L$.