query: Show that any subring $R\subseteq L$ is a field if $K\subseteq L$ is module-finite field extension and $K\subseteq R$..
positive_passages.docid: 142336
positive_passages.text: Why does the integral domain "being trapped between a finite field extension" implies that it is a field?
negative_passages.docid: 42652
negative_passages.text: Must $k$-subalgebra of $k[x]$ be finitely generated?
negative_passages.docid: 46565
negative_passages.text: Show that $\mathbb{Q}(\sqrt{2})$ is a field.
negative_passages.docid: 46648
negative_passages.text: How to prove the existence of a splitting field?
negative_passages.docid: 137858
negative_passages.text: Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
negative_passages.docid: 111444
negative_passages.text: Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
negative_passages.docid: 147200
negative_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: 119305
negative_passages.text: Is the splitting field equal to the quotient $k[x]/(f(x))$ for finite fields?
negative_passages.docid: 29635
negative_passages.text: Fields finitely generated as $\mathbb Z$-algebras are finite?
negative_passages.docid: 103654
negative_passages.text: A finite field extension that is not simple
negative_passages.docid: 7567
negative_passages.text: Degree of Field Extensions and $F(\alpha)$
negative_passages.docid: 52212
negative_passages.text: If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
negative_passages.docid: 131943
negative_passages.text: How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
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: 115129
negative_passages.text: Constructing an explicit isomorphism between finite extensions of finite fields
negative_passages.docid: 93328
negative_passages.text: $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$