Norm of a polynomial $N_{E/K}(f)(X)$ with $E/K$ a Galois extension

Question

I am quite new to Galois Theory and tried to give a go at this exercise:

Let $E/K$ be a finite Galois extension with $G=\text{Gal}(E/K)$. For each polynomial $f(X)\in E[X]$ we define: $$N_{E/K}(f)(X)=\prod_{\sigma \in G} f^{\sigma}(X)$$

1. Check that $N_{E/K}(f)(X) \in K[X]$.
2. Prove that if $N_{E/K}(f)(X)$ is irreducible in $K[X]$, then $f(X)$ is irreducible in $E[X]$. Is the converse true?
3. Prove that if $N_{E/K}(f)(X)$ is irreducible in $K[X]$ then the coefficients of $f$ generate $E$ over $K$.
4. Let $f(X)\in E[X]$ be an irreducible polynomial and let $\alpha$ be a root of this polynomial in an algebraic closure $\bar{E}$. Prove that the normal closure of the extension $E(\alpha)/K$ in $\bar{E}$ is the splitting field of the polynomial $N_{E/K}(f)(X)$.

I thought I understood the theory, but I don't know where to begin. For (1), I'd like to see that the coefficients of the polynomial are in $K$, but how do I continue? I know $\sigma$ fixes $K$ but $f(X)\in E(X)$. For the others, I lay the definitions and try to go for a contradiction but nothing comes from it.

I've looked all over the reference books and the internet and can only find similar definitions regarding elements of $E$, not polynomials. Any help, indications or references would be greatly appreciated.

Answer 1

1. $\sigma(f(X))$ means applying $\sigma$ to the coefficients, usually denoted $f^\sigma(X)$ (well KCd prefers the notation $(\sigma f)(X)$) and the subring of $E[X]$ fixed by $Gal(E/K)$ is $K[X]$.

2. is obvious

3. for $E/F/K$ then $F=E$ iff $F$ is not fixed by any $\sigma$.

4. the roots of $f^\sigma$ are $K$-conjugates to the roots of $f$.