Is the ideal $(x^2 + 3x + 7)$ maximal, prime or radical in $\mathbb{R}[x]$?
How about in $\mathbb{C}[x]$?
For $\mathbb{R}[x]$, I first want to try and show it is or is not maximal, if it is not, then show it is prime, and if not, show it is radical. My instinct is to say that it is a prime ideal, but I do not know how to go about and do this.
the discriminant of the polynomial $f=x^2+3x+7$ is $3^2-4\cdot7 = -19$. since this is not the square of any element in $\mathbb{R}$, $f$ is irreducible over $\mathbb{R}$.
as $\mathbb{R}[x]$ is a principal ideal domain, irreducibles are prime and primes are maximal. so proving this is the trickiest bit of the problem. but perhaps you are intended simply to quote these well-known results?
suppose for any $g \in \mathbb{R}[x]$ we have $g^n=hf$ for $n \ge 1$. then $f|g$ so $g=h_1f$ and $g \in (f)$, so $(f)$ is radical