"Let $R$ be an integral domain. Assuming that the Division Algorithm holds in $R[x]$, prove that $R$ is a field."
13983
Show that If $R[X]$ is Euclidean domain then $R$ is a field
76496
Showing that every field is an integral domain.
122415
How to show every field is a Euclidean Domain.
81982
Is $\mathbb{Z}[x]$ a principal ideal domain?
13174
Show that $A[X]/(aX+b)$ 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
138826
To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
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.
134458
Isomorphisms preserving integral domains and fields
95408
Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
46047
Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain
4401
Constructive proof that algebraic numbers form a field
133636
Is $\mathbb Z[X]/(X^2-1)$ a domain?
10111
Prove or disprove that Q[√2] is a field
112345
In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?
5893
On the commutative property of multiplication (domain of integers, possibly reals)
89271
If $F$ is a field, then $F[x]$ is a principal ideal domain proof question