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> Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q\_5$ of $5$-adic numbers a number field, if yes what is the degree ?
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To be honest I don't understand the question, what does it mean to be the algebraic closure of $\mathbb Q$ in the field $\mathbb Q\_5$? Maybe $\bar{\mathbb Q}\cap\mathbb Q\_5$ ?

on Wikipedia I saw also;

''The algebraic closure of a field $K$ has the same cardinality as $K$ if $K$ is infinite, and is countably infinite if $K$ is finite.''

Therefore If $\mathbb Q$ is not entirely contained in $\mathbb Q\_5$, $\bar{\mathbb Q}\cap\mathbb Q\_5$ cannot be a finite extension of $\mathbb Q$, is that correct ?