Prove that if $R[x]$ is a PID, then $R$ is a field
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
55902
Show $R[x]/(x)\cong R$
23721
Prove that the Gaussian Integer's ring is a Euclidean domain
121076
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
85096
Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
57669
A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
21747
A subring of the field of fractions of a PID is a PID as well.
4401
Constructive proof that algebraic numbers form a field
8317
Understanding a proof of "If $R$ is an integral domain, then $R[x]$ is an integral domain as well".
44259
Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)
122415
How to show every field is a Euclidean Domain.
25397
Prove $R \times R$ is NOT an integral domain
149783
$\mathbb Z\times\mathbb Z$ is principal but is not a PID
24224
How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
129943
Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$
57592
Does existence of $\gcd$ implies PID?
151803
Prove that $R$ is commutative