query: A solution of a linear system in some extension field implies a solution in the subfield
positive_passages.docid: 22192
positive_passages.text: System of linear equations having a real solution has also a rational solution.
negative_passages.docid: 152451
negative_passages.text: REVISITED $^2$: Does a solution in $\mathbb{R}^n$ imply a solution in $\mathbb{Q}^n$?
negative_passages.docid: 140212
negative_passages.text: Prove that there exists a unique linear map that extends the map from a basis
negative_passages.docid: 116955
negative_passages.text: Showing the existence of a certain kind of extension of a finitely generated algebra
negative_passages.docid: 28313
negative_passages.text: Intermediate ring between a field and an algebraic extension.
negative_passages.docid: 56551
negative_passages.text: Multiplicative group of field extension over base field
negative_passages.docid: 111122
negative_passages.text: Proof of a theorem in Hilbert's system
negative_passages.docid: 7828
negative_passages.text: Proof that an integral domain that is a finite-dimensional $F$-vector space is in fact a field
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 111347
negative_passages.text: Finding an example of field extensions satisfying certain properties
negative_passages.docid: 57283
negative_passages.text: Extending Homomorphism into Algebraically Closed Field
negative_passages.docid: 69839
negative_passages.text: Prove that a system of linear equations, will have infinitely many solutions whenever there is a specific value.
negative_passages.docid: 79450
negative_passages.text: For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$
negative_passages.docid: 142725
negative_passages.text: The general solution of $x^a = a^x$ for real $a >0$
negative_passages.docid: 146446
negative_passages.text: Infinite Degree Algebraic Field Extensions
negative_passages.docid: 101166
negative_passages.text: Show that for $\eta > 0$, $\alpha_\eta=-\infty$ and for the Cauchy problem $z'=x-z^2,z(0)=\eta$ the saturated solution is defined up to $+\infty$.
negative_passages.docid: 83009
negative_passages.text: Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field