Every element of $\mathbb{Q}[\alpha]$ is algebraic over $\mathbb{Q}$ for an $\alpha$ algebraic
74058
Sums and products of algebraic numbers
45302
Finding the $\mathbb{Q}$-automorphisms of the splitting field of $x^p-2$ over $\mathbb{Q}$.
77791
Proving an inclusion related to algebraic sets and interpreting it
38489
Degree of the extension $\mathbb{Q}(\sqrt{a+\sqrt{b}})$ over $\mathbb{Q}$
127002
Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
8087
Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer
24466
Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
55764
Algebraic closure of $\mathbb F_p$
122083
Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$
135441
When is $\mathbb{Z}[\alpha]$ dense in $\mathbb{C}$?
116955
Showing the existence of a certain kind of extension of a finitely generated algebra
129239
Additive quotient group $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the multiplicative group of roots of unity
31451
Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID
107821
The ring of integers of $\mathbf{Q}[i]$
130944
Greatest common divisor of polynomials over $\mathbb{Q}$
79679
Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.
99423
Is it true every ideal of $\mathbb{Z}[x]$ is finitely generated?