Let $F$ be a field. Could the ring $F[x]$ be a field?
46254
Why can't the Polynomial Ring be a Field?
116916
What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
51612
Question About Notation In Field Theory $F(x)$ vs. $F[x]$
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
81982
Is $\mathbb{Z}[x]$ a principal ideal domain?
49121
Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions
20218
Is this field extension finite?
155745
Ideals of $\mathbb{Z}[X]$
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
133561
show that if $K$ is a field then $K[x]$ is principal
124672
Let $f:R\rightarrow S$ be a ring homomorphism. Prove or disprove: if $f$ is onto and $R$ is an integral domain, then $S$ is an integral domain
151018
Proving that any finite integral domain is a field.
80961
Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
129597
Is a field perfect iff the primitive element theorem holds for all extensions, and what about function fields
70751
A commutative ring whose all proper ideals are prime is a field.
92677
Infinite algebraic extension of a finite field
80879
Can a quotient field ever be finitely generated as an algebra?