Why $k[X]/\langle f\rangle$ is a field?

Question

I am studying extensions of fields. $k[X]$ is a ring of polynomials (with coefficients in $k$) and $\langle f\rangle$ is an ideal generated by $f$, where $f$ is irreducible of degree $n\geq 1$.

Answer 1

$k[X]$ is a PID so if $f \in k[X]$ is irreducible, then $(f)$ is a maximal ideal. Since we have

$R/I$ is a field if and only if $I$ is a maximal ideal

then $k[X]/(f)$ is a field.