Proving $f\in K(\beta)[X]$ irreducible iff $g\in K(\alpha)[X]$ irreducible
7858
Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$
93658
Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible
99341
Basic question: closure of irreducible sets
27942
showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
143095
If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.
8977
Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
152109
In a UFD if $x$ is irreducible, then $(x)$ is prime
66355
A sufficient condition for irreducibility of a polynomial in an extension field.
27577
An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension
102871
$\langle r\rangle$ maximal $\iff r$ irreducible
91356
Irreducibility of an infinite sequence of polynomials
132665
Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
141792
Slick proof the determinant is an irreducible polynomial
35959
irreducibility of a polynomial over $\mathbb{F}_2$
140497
Irreducibles are prime in a UFD