query: $X^{p^n} - a$ is irreducible if $a \in K$ has no $p$-th root?
positive_passages.docid: 95594
positive_passages.text: Prove that a polynomial is irreducible or the field contains a $p$th root
negative_passages.docid: 51952
negative_passages.text: Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
negative_passages.docid: 1487
negative_passages.text: How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
negative_passages.docid: 116856
negative_passages.text: Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 63605
negative_passages.text: Irreducible polynomials and affine variety
negative_passages.docid: 91356
negative_passages.text: Irreducibility of an infinite sequence of polynomials
negative_passages.docid: 70537
negative_passages.text: Irreducible polynomials have distinct roots?
negative_passages.docid: 27942
negative_passages.text: showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 16035
negative_passages.text: Irreducibility of $X^{p-1} + \cdots + X+1$
negative_passages.docid: 46292
negative_passages.text: Find all primes $p>2$ for which $x^2+x+1$ is irreducible in $\mathbb{F}_p[x]$
negative_passages.docid: 132572
negative_passages.text: Why this polynomial is irreducible?
negative_passages.docid: 26069
negative_passages.text: Why is $X^4-16X^2+4$ irreducible in $\mathbb{Q}[X]$?
negative_passages.docid: 70249
negative_passages.text: Irreducibility of Polynomials in $k[x,y]$
negative_passages.docid: 4706
negative_passages.text: Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
negative_passages.docid: 93196
negative_passages.text: $x^4 -10x^2 +1 $ is irreducible over $\mathbb Q$
negative_passages.docid: 29545
negative_passages.text: $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
negative_passages.docid: 84355
negative_passages.text: Methods to see if a polynomial is irreducible