query: Prove that the polynomial $X^2 +X +1$ is reducible in $\Bbb F_p[X]$ if and only if $p ≡ 1\pmod 3.$
positive_passages.docid: 152591
positive_passages.text: For which prime numbers $p$ does the congruence $x^2 + x + 1 \equiv 0 \pmod{p}$ have solutions?
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: 152347
negative_passages.text: Proof that a certain entire function is a polynomial
negative_passages.docid: 25471
negative_passages.text: Irreducibility of $x^n-x-1$ over $\mathbb Q$
negative_passages.docid: 103552
negative_passages.text: The minimal polynomial of a primitive $p^{m}$-th root of unity over $\mathbb{Q}_p$
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: 121717
negative_passages.text: Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
negative_passages.docid: 120627
negative_passages.text: Reducibility of polynomials modulo p
negative_passages.docid: 16592
negative_passages.text: Irreducible polynomial which is reducible modulo every prime
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: 70523
negative_passages.text: Proving that $x^4 - 10x^2 + 1$ is not irreducible over $\mathbb{Z}_p$ for any prime $p$.
negative_passages.docid: 151985
negative_passages.text: If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
negative_passages.docid: 93328
negative_passages.text: $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
negative_passages.docid: 154360
negative_passages.text: Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.
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: 74777
negative_passages.text: Solvability and reducibility of a polynomial in a "chain" of finite fields
negative_passages.docid: 1962
negative_passages.text: Polynomial $P(a)=b,P(b)=c,P(c)=a$