If $F$ is a finite domain, it's a field.
551
Why is a finite integral domain always field?
137858
Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
80961
Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
89271
If $F$ is a field, then $F[x]$ is a principal ideal domain proof question
21079
Why must a field whose a group of units is cyclic be finite?
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 $K[x]$.
117005
Rules in the ring with coefficients in finite field $\mathbb{F}_p$
55492
Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$
23242
Commutative integral domain with d.c.c. is a field
8317
Understanding a proof of "If $R$ is an integral domain, then $R[x]$ is an integral domain as well".
23712
$F$ is a finite field of size $s$. Prove that if $s=2^m$ for some $m>0$, then all elements of F have a square root.
112909
is a number field by definition a subfield of $ \mathbb C $?
62757
Showing that a ring is a field as well for one of the provided choices.
19671
Does the "field" over which a vector space is defined have to be a Field?
7011
Integral domain with fraction field equal to $\mathbb{R}$
51612
Question About Notation In Field Theory $F(x)$ vs. $F[x]$
31961
Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$?