Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.
34754
Why is the polynomial ring of more than one variable not a PID?
154360
Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.
144572
Proof that ideals in $C[0,1]$ are of the form $M_c$ that should not involve Zorn's Lemma
79041
What is the formal definition of polynomial ring of several variables?
82485
Show that the set of non zero rationals is not a cyclic group under multiplication.
85633
Rings of polynomials whose (partial) derivatives vanish are Noetherian?
82771
Proving that any polynomial $p:\mathbb{R}\rightarrow \mathbb{R}$ is not bounded, but its restriction to any bounded set is bounded
98577
Show that the set $\mathbb{Q}(\sqrt{p})=\{a+b\sqrt{p}; a,b,p\in\mathbb{Q},\sqrt{p}\notin \mathbb{Q}\}$ is a field
78873
How to show that the monomials are not a Schauder basis for $C[0,1]$
78950
Does this map extend to a unique automorphism of the polynomial ring of an integral domain?
46254
Why can't the Polynomial Ring be a Field?
156034
sufficient condition for a polynomial to have roots in $[0,1]$
145879
$R$ is PID, so $R/I$ is PID, and application on $\mathbb{Z}$ and $\mathbb{N}$
75558
Counter-example for prime ideals in a non commutative ring.
121845
Quotient of polynomials, PID but not Euclidean domain?
65637
Squares in Polynomial Quotient Ring $ (\mathbb{Z}/2)[x] /(x^{17} - 1) $
5802
Proving the universal mapping property for polynomial rings