F[x] is Principal Ideal Domain implies F is a Field. Note: This is the converse of a known theorem.
33576
Non-principal ideal in $K[x,y]$?
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
15137
Does $IJ=IK\implies J=K$ always hold for integral domain and finitely generated nonzero ideal $I$?
141659
Every field is an integral domain
150239
In an integral domain every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.
46047
Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain
8317
Understanding a proof of "If $R$ is an integral domain, then $R[x]$ is an integral domain as well".
93621
Prove that $\mathbb{Z}[\omega]$ is an integral domain
90533
Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.
33586
Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
109691
When are kernels of homomorphisms on $\mathbb{Z}[x]$ principal ideals, and what does this have to do with Gauss's lemma?
151794
Can an ideal in a commutative integral domain be its own square?
22789
How to determine whether a unique factorization domain is a principal ideal domain?
76661
Checking the maximality of a principal ideal in $R[x]$
155745
Ideals of $\mathbb{Z}[X]$
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
121986
Do the Laurent polynomials over $\mathbb{Z}$ form a principal ideal domain?