query: If $X^p-a$ has no zeros in a field $F$ of characteristic $p$ where $a \in F$, is it irreducible?
positive_passages.docid: 18336
positive_passages.text: $x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?
negative_passages.docid: 54192
negative_passages.text: Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
negative_passages.docid: 121663
negative_passages.text: If $a$ is a zero divisor of $\mathbb{F}$, then does $a$ fail to have a multiplicative inverse?
negative_passages.docid: 120940
negative_passages.text: An irreducible polynomial in a subfield of $\mathbb{C}$ has no multiple roots in $\mathbb{C}$
negative_passages.docid: 46951
negative_passages.text: For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$?
negative_passages.docid: 115782
negative_passages.text: Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
negative_passages.docid: 154373
negative_passages.text: How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p[X]$?
negative_passages.docid: 27942
negative_passages.text: showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 102422
negative_passages.text: Irreducibly of $x^5-x-1$ over $\mathbb{F}_3$
negative_passages.docid: 38364
negative_passages.text: Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
negative_passages.docid: 140808
negative_passages.text: Problem on Galois theory and irreducible polynomial
negative_passages.docid: 65558
negative_passages.text: Irreducible in $\mathbb{Z}[\sqrt{-5}]$
negative_passages.docid: 70537
negative_passages.text: Irreducible polynomials have distinct roots?
negative_passages.docid: 127002
negative_passages.text: Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
negative_passages.docid: 90533
negative_passages.text: Show that if $p(x)$ is reducible in $F[x]$, then $F[x]/(p(x))$ is not an integral domain.
negative_passages.docid: 156671
negative_passages.text: Necessary and sufficient condition for $x^n - y^m$ to be irreducible in $\Bbb C[x,y]$
negative_passages.docid: 70523
negative_passages.text: Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.