Galois group of $f$ is cyclic if $\deg f$ is prime

Question

Hello I am learning Galois Theory by myself and got lost in the following exercise:

Let $f$ be an irreducible polynomial of degree $n$, and suppose that the splitting field of $f$ is generated by a single root of $f$.

a) Show that the Galois group of $f$ is cyclic if $n$ is prime, and that it is either cyclic or a Klein 4-group, if $n = 4$.

b) Let $K$ be any Galois extension of $\mathbb Q$ of degree $n$. Show that there exists an irreducible polynomial $f$ of degree $n$ having $K$ as its splitting field, and that $K$ can be generated by a single root of this polynomial.

Thanks

Answer 1

a) Hint: The order of the Galois group of a polynomial is the same as the degree of its splitting field. Furthermore, any group of prime order is cyclic. A group of order $4$ is abelian, and is either cyclic or isomorphic to Klein's Vierergruppe.

b) Hint: Think of the Theorem of the Primitive element.