$x^p-a$ either has a root or is irreducible
18336
$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?
155962
Are there irreducible polynomials that are partially solvable by radicals?
70249
Irreducibility of Polynomials in $k[x,y]$
16035
Irreducibility of $X^{p-1} + \cdots + X+1$
121076
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
27577
An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension
106914
Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree
84355
Methods to see if a polynomial is irreducible
132665
Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
75528
$x^2-x+1$ has a root $\!\bmod p\,$ for infinitely many primes $p$
27942
showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
90612
Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots
93196
$x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$
25471
Irreducibility of $x^n-x-1$ over $\mathbb Q$
118950
Irreducibility criteria for polynomials with several variables.
86038
Family of irreducible polynomials
78143
Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements