if $a$ is algebraic, then $k[a] = k(a)$
52212
If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
109745
If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.
63390
Infinite algebraic extensions of $\mathbb{Q}$
464
Prove that the ideal $(X_1-a_1,...,X_n-a_n)$ is maximal in $K[X_1,\dots,X_n]$
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
151806
Prove that $A^k = 0 $ iff $A^2 = 0$
6896
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
24551
How to prove that the sum and product of two algebraic numbers is algebraic?
8587
What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?
62687
Proving Cardinality of Cosets in Abstract Algebra
120933
A is similar to $A^k$, then each eigenvalue of $A$ is a root of unity
122083
Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$
106584
If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
58424
If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field.
120877
$A$ abelian variety. Is the multiplication by $n_A$ surjective?
56171
Is $K(X)$ never algebraically closed?
126427
Proving $\sqrt{2}+\sqrt[3]{3}$ is algebraic.