Let $F$ be a field. How do we show that maximal ideals of $F[x]$ are the principal ideals generated by the monic irreducible polynomials? Let $F$ be a field. How do we show that maximal ideals of $F[x]$ are the principal ideals generated by the monic irreducible polynomials?




In Algebra by Artin, he says this proposition is proven analogously to: 


  


Here, he shows that if $n$ is prime, then $\mathbb Z/(n)$ is a field. Then we use the fact that $R/I$ is a field iff $I$ is maximal, and he concludes that $(n)$ is maximal. 



The analogous proof would be that if $f(x)$ is monic irreducible, then $F[x]/(f)$ is a field. The only problem is that he has not proven that $F[x]$ modulo a monic irreducible polynomial is a field.
For any field $F$, $F[x]$ is a principal ideal domain; this is a very well-known and oft-quoted result, which I will accept here.
Now let
$M \subset F[x] \tag 1$
be a maximal ideal; since $F[x]$ is a principal ideal domain, we have
$M = (m(x)) \tag 2$
for some
$m(x) \in F[x]; \tag 3$
we may clearly take $m(x)$ to be monic, since the leading coefficient $\mu$ of $m(x)$, satisfying as it does $\mu \ne 0$, is a unit; thus $\mu^{-1} m(x)$ is monic and
$(\mu^{-1} m(x)) = (m(x)); \tag 4$
now if $m(x)$ were reducible in $F[x]$, we would have
$m(x) = p(x)q(x), \; p(x), q(x) \in F[x], \; \deg p(x), \deg q(x) \ge 1; \tag 5$
consider the ideal
$(p(x)) \subsetneq F[x]; \tag 6$
it is clearly proper:  since $\deg p(x) \ge 1$, $(p(x))$ contains no polynomials of degree $0$, that is, contains no elements of $F$ other than $0$. Also,
$(m(x)) = (p(x)q(x)) \subsetneq (p(x)), \tag 7$
for
$p(x) \notin (p(x)q(x)) \tag 8$
lest for some
$r(x) \in F[x] \tag 9$
we have
$p(x) = r(x)p(x)q(x), \tag{10}$
or
$p(x)(r(x)q(x) - 1) = 0, \tag{11}$
whence
$r(x)q(x) = 1, \tag{12}$
which yields
$\deg r(x) + \deg q(x) = \deg 1 = 0, \tag{13}$
impossible in light of the assumption $\deg q(x) \ge 1$; thus we have shown that
$(m(x)) = (p(x)q(x)) \subsetneq (p(x)) \subsetneq F[x] \tag{14}$
in the event that $m(x)$ is reducible, which further shows that $(m(x))$ is not a maximal ideal in $F[x]$; this contradiction implies that $m(x)$ is irreducible in $F[x]$.  Finis.
There's a classic result in commutative algebra that you can apply:

Let $B$ be an integral domain, $A$ be a subring such that $B$ is
integral over $A$. Then $B$ is a field if and only if $A$ is a field.