Define the elementary symmetric functions in the variables $x_{1}, \ldots, x_{n}$. State the fundamental theorem of symmetric functions.

Let $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \in K[x]$, where $K$ is a field. Define the discriminant of $f$, and explain why it is a polynomial in $a_{0}, \ldots, a_{n-1}$.

Compute the discriminant of $x^{5}+q$.

Let $f(x)=x^{5}+p x^{2}+q$. When does the discriminant of $f(x)$ equal zero? Compute the discriminant of $f(x)$.