Special polynomials in $\mathbb{Z}[X]$ have no roots in $\mathbb{Z}$ and $\mathbb{Q}$
30511
Monic $f(x)\in\Bbb Z[x]$ has no rational root if $f(0)\ \&\ f(1)$ odd [Parity Root Test, Modular Root Test]
94682
Polynomial: prove there is no double root
155745
Ideals of $\mathbb{Z}[X]$
39346
Let $R= \mathbb{Q}[x^2 ,x^3 ]$, the set of all polynomials over $\mathbb{Q}$ with no $x$ term. Show that the $\gcd$ of $x^5$ and $x^6$ does not exist
10514
Proof that no polynomial with integer coefficients can only produce primes
95146
Polynomial ring $R[X] $ has zero Jacobson radical, for $R$ a domain
31216
Is the image of $\Phi_n(x) \in \mathbb{Z}[x]$ in $\mathbb{F}_q[x]$ still a cyclotomic polynomial?
99835
Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
77820
Roots in a polynomial ring
85633
Rings of polynomials whose (partial) derivatives vanish are Noetherian?
21862
Proof of lack of pure prime producing polynomials.
105679
Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients?
92703
Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.
134342
Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit.
53012
Irreducibility of a polynomial if it has no root (Capelli)
64878
Is this polynomial irreducible in $\mathbb{Q}$?
8087
Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer