query: Non-Relatively Prime Polynomials in a Field Extension are Not Relatively Prime in Original Field Either
positive_passages.docid: 147200
positive_passages.text: Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$?
negative_passages.docid: 79450
negative_passages.text: For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$
negative_passages.docid: 119545
negative_passages.text: Nonzero rationals under multiplication are not a cyclic group
negative_passages.docid: 79226
negative_passages.text: Discrete non archimedean valued field with infinite residue field
negative_passages.docid: 4192
negative_passages.text: Non-algebraically closed field in which every polynomial of degree $<n$ has a root
negative_passages.docid: 83595
negative_passages.text: Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
negative_passages.docid: 87101
negative_passages.text: Irreducible but not prime in $K[X,Y]/(X^2-Y^3)$
negative_passages.docid: 158924
negative_passages.text: Example of composition of two normal field extensions which is not normal.
negative_passages.docid: 117206
negative_passages.text: $|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
negative_passages.docid: 25229
negative_passages.text: Can any set of $n$ relatively prime elements be extended to an invertible matrix?
negative_passages.docid: 7452
negative_passages.text: Splitting of primes in the compositum of fields
negative_passages.docid: 133975
negative_passages.text: Localising a polynomial ring and non-maximal prime ideal
negative_passages.docid: 17991
negative_passages.text: Nonconstant polynomials do not generate maximal ideals in $\mathbb Z[x]$
negative_passages.docid: 21862
negative_passages.text: Proof of lack of pure prime producing polynomials.
negative_passages.docid: 132423
negative_passages.text: Non trivial valuation over a finite extension $F/\mathbb{Q}_p$ is equivalent to the one induced by $v_p$
negative_passages.docid: 139463
negative_passages.text: Commutative integral domain does not finitely generate its field of fractions
negative_passages.docid: 68753
negative_passages.text: Elements in finite field extensions