(a) Let $K$ be a field. Define the discriminant $\Delta(f)$ of a polynomial $f(x) \in K[x]$, and explain why it is in $K$, carefully stating any theorems you use.

Compute the discriminant of $x^{4}+r x+s$.

(b) Let $K$ be a field and let $f(x) \in K[x]$ be a quartic polynomial with roots $\alpha_{1}, \ldots, \alpha_{4}$ such that $\alpha_{1}+\cdots+\alpha_{4}=0$.

Define the resolvant cubic $g(x)$ of $f(x)$.

Suppose that $\Delta(f)$ is a square in $K$. Prove that the resolvant cubic is irreducible if and only if $G a l(f)=A_{4}$. Determine the possible Galois groups Gal $(f)$ if $g(x)$ is reducible.

The resolvant cubic of $x^{4}+r x+s$ is $x^{3}-4 s x-r^{2}$. Using this, or otherwise, determine $\operatorname{Gal}(f)$, where $f(x)=x^{4}+8 x+12 \in \mathbb{Q}[x]$. [You may use without proof that $f$ is irreducible.]