This is one of my mid-term problems:
Let $E$ be the set of all complex elements of $\mathbb{C}$ over $\mathbb{Q}$, i.e, $E$ = {u $\in$ $\mathbb{C}$|u is algebraic over $\mathbb{Q}$}. Prove that:
a) $E$ is an algebraic and infinite extension of $\mathbb{Q}$.
b) $E$ is a normal and seperable extension of $\mathbb{Q}$.
My answer for this problem is as follow:
a) $E$ is clearly an algebraic extension since all of its elements are algebraic.
Next, we suppose that E is finite, that means E is a finite vector space over $\mathbb{Q}$.
Assume that [$E$:$\mathbb{Q}$] = n, then it has a basis of {$\alpha_{1}$;$\alpha_{2}$;...;$\alpha_{n}$}
b) For normality, I attempt to show that if a polynomial p(x) has root $u$ in $E$, then it's other roots {$u_{1};u_{2};...;u_{n}$} are all so in $E$, but how to do that then I have no idea.
In terms of seperability, I'm using a result:
If char $E$ = 0 then every irreducible polynomial in $E[x]$ is seperable
Can I get any hint for this problem please?