Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
53012
Irreducibility of a polynomial if it has no root (Capelli)
115299
Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$
151961
Irreducibility of $x^{p(p-1)}+x^{p(p-2)}+\cdots+x^p+1$ over $\mathbb Q$
38364
Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
121076
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
116856
Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$
128043
$A^n=0$ for $A$ nilpotent without the use of minimal polynomial
8494
Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.
83291
Prove that $p(x)=(x-a)q(x)$ for some $q(x)\in R[x]$ of degree= (degree of $p)-1$.
121301
What are the roots of an irreducible $p\in F[X]$ in the field extension $F[X]/\langle p\rangle$?
10627
When is $X^n-a$ is irreducible over F?
97729
Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible
137233
$f$ is irreducible $\iff$ $G$ act transitively on the roots
154373
How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p[X]$?
78143
Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements
41693
Use irreducible fibers to show $X$ is irreducible
112427
Irreducibility of $p(x)=x^4-4x^2+8x+2$ over $\mathbb{Q}(\sqrt{-2})$- Dummit Foote Abstract algebra $9.4.10$