$E$ be an algebraic extension of $F$. Suppose $K$ is an integral domain s.t. $F \leq K \leq E$. Show that $K$ is a field.
128550
Algebraic field extension and intermediate integral domain
22544
Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
78950
Does this map extend to a unique automorphism of the polynomial ring of an integral domain?
57283
Extending Homomorphism into Algebraically Closed Field
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
87103
Quick question about ideals of field $F$ in $F[x]$ are principal.
111444
Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
158525
Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?
48873
Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.
94579
$\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$
141659
Every field is an integral domain
62134
Show that if $ R $ is an integral domain then a polynomial in $ R[X] $ of degree $ d $ can have at most $ d $ roots
131943
How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
91622
Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.
52212
If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
106737
A problem on field extension
74663
Is $\mathbb{R}$ an algebraic extension of some proper subfield?