query: Let $a \in F - F^{p}$. Show that $x^{p} - a$ is irreducible over $F$.
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: 95594
negative_passages.text: Prove that a polynomial is irreducible or the field contains a $p$th root
negative_passages.docid: 115782
negative_passages.text: Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
negative_passages.docid: 127310
negative_passages.text: Suppose $\gcd(\deg(f),\deg (g))=1$. Show that $g(x)$ is irreducible in $k(\alpha)[X]$.
negative_passages.docid: 96087
negative_passages.text: Let $\mathbb{F}_2 \cong \mathbb{Z}/2\mathbb{Z}$. Is $x^4+x^2+1$ irreducible in $\mathbb{F}_2[x]$?
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: 76640
negative_passages.text: Show that the polynomial $x^3-xy^2+y+1$ is irreducible in $\mathbb{Q}[x,y]$
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: 6722
negative_passages.text: to show $f$ is a polynomial in $z$
negative_passages.docid: 38364
negative_passages.text: Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
negative_passages.docid: 120627
negative_passages.text: Reducibility of polynomials modulo p
negative_passages.docid: 32391
negative_passages.text: Show that $x^2+x+4$ is irreducible over $\mathbb Z_{11}$.
negative_passages.docid: 35959
negative_passages.text: irreducibility of a polynomial over $\mathbb{F}_2$
negative_passages.docid: 97729
negative_passages.text: Show that a polynomial $f(x)$ over a field $k$ is irreducible if and only if the polynomial $f(x+1)$ is irreducible
negative_passages.docid: 8977
negative_passages.text: Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 19968
negative_passages.text: Describe all $p^{n}$ (in terms of congruence conditions of $p$ and $n$) for which $x^{2}+1$ irreducible over $\mathbb{F}_{p^{n}}$.