query: $K$ finite field, $P'(X) = 0$ then P is reductible
positive_passages.docid: 154360
positive_passages.text: Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.
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: 77620
negative_passages.text: Irreducible polynomials over $\mathbb Q$ and $\mathbb Z $
negative_passages.docid: 138083
negative_passages.text: $\gcd(X^n - 1, nX^{n-1}) = 1$ in a field $K$ of characteristic $0$ or characteristic $p \nmid n$.
negative_passages.docid: 1487
negative_passages.text: How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?
negative_passages.docid: 85096
negative_passages.text: Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.
negative_passages.docid: 140106
negative_passages.text: How to prove a finite field is not ordered?
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: 134726
negative_passages.text: Show the polynomial is irreducible in $\textbf Z_2$
negative_passages.docid: 25187
negative_passages.text: Why $∀x(P(x)→Q)$ is equivalent to $∃xP(x)→Q$
negative_passages.docid: 112427
negative_passages.text: Irreducibility of $p(x)=x^4-4x^2+8x+2$ over $\mathbb{Q}(\sqrt{-2})$- Dummit Foote Abstract algebra $9.4.10$
negative_passages.docid: 133561
negative_passages.text: show that if $K$ is a field then $K[x]$ is principal
negative_passages.docid: 86174
negative_passages.text: Finding an isomorphism between these two finite fields
negative_passages.docid: 29545
negative_passages.text: $[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
negative_passages.docid: 48093
negative_passages.text: Irreducible polynomial in field extension
negative_passages.docid: 38364
negative_passages.text: Showing that $x^n -2$ is irreducible in $\mathbb{Q}[X]$
negative_passages.docid: 134701
negative_passages.text: Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$