# Questions tagged [algebraic-integers]

For questions regarding algebraic integers, which is a complex number which is integral over the integers.

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### Problem relating $p$-defect zero characters with their values on a field of characteristic $p$

Studying Gabriel Navarro's book "Character Theory and the McKay Conjecture", I've come across the following problem. First, let's fix some notation: $G$ will be a finite group, $R$ will ...
• 2,629
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### The house of an algebraic number

Let $\alpha$ be a non-zero algebraic number of degree $d$. Denote ${\rm den}(\alpha)$ the smallest positive integer $m$ such that $m\alpha$ is an algebraic integer, and ${\rm House}(\alpha)$ the ...
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### Sums of powers and algebraic integers

Suppose that $x_1, \dots, x_n$ are algebraic over $\mathbb{Q}$, and for every integer $N\geq 1$, the sum $\sum_{i=1}^{n}x_i^N$ is an algebraic integer. Does this imply that $x_1,\dots,x_n$ are also ...
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• 509
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### Upper bound on number of algebraic integers of degree $\leq d$

Let $\alpha \in \mathbb{C}$ be an algebraic integer, which means that it has a monic polynomial $f=X^d + a_1X^{d-1} + \dots + a_d\in \mathbb{Z}[X]$ such that $f(\alpha)=0$. Over $\bar{\mathbb{Q}}$ ...
1 vote
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### Product of almost all Galois conjugates

I'm trying to prove the following: Given a matrix $M \in \mathbb{Z}^{n\times n}$ with an irreducible characteristic polynomial $f$ (irreducible over $\mathbb{Z}$ or $\mathbb{Q}$). If I'm not mistaken, ...
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1 vote
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### Proof that $\frac{V_n(k_1,...,k_n)}{V_n(1,...,n)}$ is a integer.

Let $k_1 < k_2 < ... < k_n$ will be integers. Prove that the quotient $\frac{V_n(k_1,...,k_n)}{V_n(1,...,n)}$ is a integer. where $V_n(x_1,...,x_n)$ is vandermonde determinant My idea: Show ...
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