Let $G$ be a field, $c \in G$, and let $H = \{(x – c)k, k \in G[X]\}$ be an ideal of the polynomial ring $G[X]$. How to show that $H$ is a maximal ideal?
Ideas:
I know that $H$ is maximal if and only if $H = <p(x)>$ for some irreducible polynomial $p(x)$ in $G[X]$. I also know that it suffices to show that $G[X]/H$ is a field however, I'm having trouble coming up with a coherent proof. Any help would be much appreciated.
Best Answer
Both approaches work. One way is to see that $X - c$ is irreducible: if it weren't then we would have $X-c = fg$ with $f,g$ non units and non-zero (because $X -c \neq 0$). Thus both $f$ and $g$ must have degree at least $1$, which is absurd since
$$ 1 = \deg (X-c) = \deg fg = \deg f + \deg g \geq 2 $$
The other approach is proving that $G[X]/\langle X-c\rangle$ is a field. Here's a hint: consider the unital ring morphism $G[X] \to G$ that sends $X$ to $c$, and recall the statement of the first isomorphism theorem.