How to prove that the sum and product of algebraic integers is an algebraic integer?
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Constructive proof that algebraic numbers form a field
69439
Infinite Sum of algebraic expression
62792
Can sum of a rational number and its reciprocal be an integer?
101899
Proof By Induction of a Sum
99032
Prove that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
142666
Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
79679
Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.
41486
If $x^n + a_{n-1}x^{n-1} + ... + a_0 = 0$ has a rational root, then such root is an integer
50114
A question about direct sums and products of modules
89175
Proof of $(a+b)^{n+1}$
89496
$a$ algebraic number, $\exists$ integer $n>0$ such that $na$ algebraic integer
13147
Proof verification - Minimal polynomial of $\alpha$ has integer coef.s iff its an alg. integer
14815
How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
30136
Proof that a Combination is an integer
114796
Direct proof that the product of odd integers is odd
18465
Sum and product of two transcendental numbers can't be both algebraic