query: Show that $1+X^4$ is reducible over $\mathbb Z_p$ for every prime $p$.
positive_passages.docid: 16592
positive_passages.text: Irreducible polynomial which is reducible modulo every prime
negative_passages.docid: 27942
negative_passages.text: showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 105442
negative_passages.text: Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
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: 114489
negative_passages.text: Solving $x^p + y^p = p^z$ in positive integers $x,y,z$ and a prime $p$
negative_passages.docid: 97258
negative_passages.text: Give $\ker(\phi)$ for $\phi: \mathbb{Z} \to \mathbb{Z_7}$ such that $\phi(1)=4$
negative_passages.docid: 344
negative_passages.text: Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 157884
negative_passages.text: For any prime $p>3$ show that 3 divides $2p^2+1$
negative_passages.docid: 32762
negative_passages.text: Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
negative_passages.docid: 154305
negative_passages.text: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
negative_passages.docid: 93898
negative_passages.text: Prove that for any prime p, there is a complete set of mod p residues {q1, . . . , qp}, where each qi is prime.
negative_passages.docid: 121890
negative_passages.text: Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
negative_passages.docid: 127901
negative_passages.text: If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
negative_passages.docid: 90144
negative_passages.text: Show that if $a+bi$ is irreducible over $\mathbb{Z} [i]$ then $a^2 + b^2$ is prime over $\mathbb{Z} $
negative_passages.docid: 134726
negative_passages.text: Show the polynomial is irreducible in $\textbf Z_2$
negative_passages.docid: 158182
negative_passages.text: Let $R = \mathbb{Z} + x\mathbb{Q}[x]$. Find all the irreducibles in $R$.