Prove that if $\alpha$ is an algebraic number and a root of $f(x)$ where $f$ has leading coefficient $a$, then $a\alpha$ is an algebraic integer.
89496
$a$ algebraic number, $\exists$ integer $n>0$ such that $na$ algebraic integer
41312
Question on primitive roots of unity
78561
Algebraic field extensions: Why $k(\alpha)=k[\alpha]$.
39082
Proving $x^\alpha-\alpha x \le 1- \alpha $
120629
Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them
111287
number of prime ideals in algebraic integers
8494
Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.
57639
Show there exists an $\alpha\in F_{p^n}$ such that $\alpha +\alpha^p+\alpha^{p^2}+\cdots+\alpha^{p^{n-1}}\neq 0$
52212
If $\alpha$ is an algebraic element and $L$ a field, does the polynomial ring $L[\alpha]$ is also a field?
125096
Minimal polynomial of $\alpha=\sqrt 2e^\frac{2\pi i}{3}$
37529
If $\sqrt[3]{a} + \sqrt[3]{b}$ is rational then prove $\sqrt[3]{a}$ and $\sqrt[3]{b}$ are rational
104643
Why do we use this definition of "algebraic integer"?
47949
If $a$ and $b$ are two roots of $x^4 + x^3 - 1 = 0$ prove that $ab$ is a root of $x^6 + x^4 + x^3 - x^2 - 1$.
35478
Polynomial $p(a) = 1$, why does it have at most 2 integer roots?
155165
Prove that $\mathbb F_8=\mathbb F_2[X]/(X^3+X+1)$
136770
Any field that contains the integers contains the rationals as a subfield (Analysis proof)
65252
Average of sum of unit roots is an algebraic integer