query: How can one go from "no roots" to "is irreducible" in this case?
positive_passages.docid: 146591
positive_passages.text: Inseparable polynomial over a non-perfect field
negative_passages.docid: 48093
negative_passages.text: Irreducible polynomial in field extension
negative_passages.docid: 121076
negative_passages.text: Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
negative_passages.docid: 100573
negative_passages.text: Show that $x^4-10x^2+1$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 66355
negative_passages.text: A sufficient condition for irreducibility of a polynomial in an extension field.
negative_passages.docid: 152109
negative_passages.text: In a UFD if $x$ is irreducible, then $(x)$ is prime
negative_passages.docid: 28079
negative_passages.text: Do these definitions of irreducibility of algebraic sets coincide?
negative_passages.docid: 121658
negative_passages.text: $x^2+y^2-1$ irreducible in $\mathbb{Q}[x,y]$?
negative_passages.docid: 108376
negative_passages.text: Best way to "negate" an instanceof
negative_passages.docid: 58382
negative_passages.text: Open subscheme of an irreducible component remains open?
negative_passages.docid: 84355
negative_passages.text: Methods to see if a polynomial is irreducible
negative_passages.docid: 16035
negative_passages.text: Irreducibility of $X^{p-1} + \cdots + X+1$
negative_passages.docid: 121301
negative_passages.text: What are the roots of an irreducible $p\in F[X]$ in the field extension $F[X]/\langle p\rangle$?
negative_passages.docid: 79564
negative_passages.text: Algorithm to find the irreducible polynomial
negative_passages.docid: 20612
negative_passages.text: Is fibre product of varieties irreducible (integral)?
negative_passages.docid: 102871
negative_passages.text: $\langle r\rangle$ maximal $\iff r$ irreducible
negative_passages.docid: 46891
negative_passages.text: Irreducibility of reciprocal polynomials