How to prove $\mathbb{R}[x]/(x^2 + \pi x + 2)$ is a field?

Question

I am confused about how to prove that $\mathbb{R}[x]/(x^2 + \pi x + 2)$ is a field. How would you go about doing that? Thanks

Answer 1

Hint: For any field $F$, $F[X] / (P(X))$ is a field if and only if $P(X)$ is irreducible over $F$.

Your polynomial is quadratic, a quadratic is irreducible over a field exactly when it doesn't have roots in the field.

Answer 2

Actually, the ring $\mathbb{R}[x]/(x^2+\pi{x}+2)$ is not a field. This is because the discriminant of the quadratic is $\pi^2-8$, which is a positive real number.