If $F$ is a field then every ideal in $F[x]$ is a principal ideal.
98540
If $F[x]$ is a principal domain does $F$ have to be necessarily a field?
129597
Is a field perfect iff the primitive element theorem holds for all extensions, and what about function fields
92071
A maximal ideal is always a prime ideal?
745
$F(x)$ versus $F[x]$ where $F$ is a field
36150
Ideal in a ring of continuous functions
464
Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
75899
Principal ideal domain with finitely many ideals
111114
Prove that an ideal in $k[x,y]$ is a prime ideal
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
83050
Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
99423
Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated?
87078
Prove the following set is an ideal of the ring $Z[x]$
108380
Assume $P$ is a prime ideal s.t. $K \subset P$, show $f(P)$ is a prime ideal
78540
Every irreducible polynomial f over perfect field F is separable
92519
ideal behind Maximal ideals are Prime ideals
63447
Prove that the kernel of a homomorphism is a principal ideal. (Artin, Exercise 9.13)
155991
Is it the case that every quotient ring ${F[x]}/{(p(x))}$ is a PID?