query: Show the polynomial $[(x - a_1) \cdots (x - a_n)]^4 + 1$ is irreducible
positive_passages.docid: 53669
positive_passages.text: Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$
negative_passages.docid: 62254
negative_passages.text: Show $F_{n+1} \cdot F_{n-1} = F_n^2 + (-1)^n$ for all $n \in \mathbb{N}$
negative_passages.docid: 33011
negative_passages.text: Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
negative_passages.docid: 48093
negative_passages.text: Irreducible polynomial in field extension
negative_passages.docid: 121714
negative_passages.text: Show that the polynomial $X^5 + X^3 + \bar{1}$ in $(\mathbb{Z}/2\mathbb{Z}[X])$ is irreducible
negative_passages.docid: 10821
negative_passages.text: Using Eisenstein criterion show that $x^3+x^2-2x-1$ is irreducible?
negative_passages.docid: 114762
negative_passages.text: Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.
negative_passages.docid: 76017
negative_passages.text: Prove that $x^3-2$ and $x^3-3$ are irreducible over $\Bbb{Q}(i)$
negative_passages.docid: 52528
negative_passages.text: Showing that minimal polynomial has the same irreducible factors as characteristic polynomial
negative_passages.docid: 28658
negative_passages.text: $R$ with an upper bound for degrees of irreducibles in $R[x]$
negative_passages.docid: 70249
negative_passages.text: Irreducibility of Polynomials in $k[x,y]$
negative_passages.docid: 114198
negative_passages.text: Showing that $y$ is irreducible in $\mathbb R[x,y]/(x^2+y^2-1)$.
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: 27577
negative_passages.text: An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension
negative_passages.docid: 133312
negative_passages.text: Are these polynomials irreducible in $\mathbb{Q}[x]$?
negative_passages.docid: 90612
negative_passages.text: Show that $9+9x+3x^3+6x^4+3x^5+x^6$ is irreducible given one of its roots
negative_passages.docid: 154360
negative_passages.text: Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.