If $f(x+a)$ is irreducible over $F$ then $f(x)$ is irreducible over $F$
132665
Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
39351
$f(X)$ is uncountable and hence $X$ is uncountable.
78143
Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements
80188
There are infinitely many irreducible polynomials in $\Bbb{F}_p[X]$
12596
Show that $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in R[x]$ is nilpotent iff $a_0,a_1,a_2,\ldots,a_n$ are nilpotent
250
An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root?
10627
When is $X^n-a$ is irreducible over F?
81921
If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
4706
Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
80421
Show that $a - b \mid f(a) - f(b)$
97729
Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
114198
Showing that $y$ is irreducible in $\mathbb R[x,y]/(x^2+y^2-1)$.
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
127310
Suppose $\gcd(\deg(f),\deg (g))=1$. Show that $g(x)$ is irreducible in $k(\alpha)[X]$.
140497
Irreducibles are prime in a UFD
81377
Show that $x^n + x + 3$ is irreducible for all $n \geq 2.$