If $F[x]$ is a principal domain does $F$ have to be necessarily a field?
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
74663
Is $\mathbb{R}$ an algebraic extension of some proper subfield?
33576
Non-principal ideal in $K[x,y]$?
128550
Algebraic field extension and intermediate integral domain
57669
A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
141659
Every field is an integral domain
142336
Why does the integral domain "being trapped between a finite field extension" implies that it is a field?
133561
show that if $K$ is a field then $K[x]$ is principal
16984
Finite extension of domains : preimage of non-zero ideal is non-zero?
155991
Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?
122415
How to show every field is a Euclidean Domain.
123024
A Noetherian integral domain is a UFD iff $(f):(g)$ is principal
93328
$\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
134458
Isomorphisms preserving integral domains and fields
13983
Show that If $R[X]$ is Euclidean domain then $R$ is a field
76661
Checking the maximality of a principal ideal in $R[x]$