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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 ...
Gauss's user avatar
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4 votes
1 answer
218 views

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 ...
Mystery girl's user avatar
0 votes
0 answers
38 views

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 ...
Absol's user avatar
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2 votes
0 answers
51 views

Find all units in the ring $\Bbb Z[\omega]$ where $\omega$ is the primitive $p^{th}$ root of unity.

Let $p>0$ be a prime and let $\omega$ be a primitive $p^{\text{th}}$ root of unity. I am trying to find all units in the ring $\Bbb Z[\omega]$. Every element of $\Bbb Z[\omega]$ is of the form $z=...
user108580's user avatar
1 vote
1 answer
76 views

The notion of "units" is independent of the base algebraic integer ring. [duplicate]

Consider the ring $\mathcal O$ of all algebraic integers and a subring $\mathcal A\subset \mathcal O$. If $u\in\mathcal A$ is invertible in $\mathcal O$, then is the inverse of $u$ necessarily in $\...
user108580's user avatar
2 votes
1 answer
111 views

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 $\bar{\mathbb{Q}}$ ...
MarvinsSister's user avatar
1 vote
0 answers
75 views

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, ...
MatthysJ's user avatar
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3 votes
1 answer
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On the proof of: If $0<\frac{|χ(g)|}{χ(1)}<1$ then $\frac{χ(g)}{χ(1)}\notin\overline{\mathbb{Z}}$

In class we proved the following theorem: Let $σ:G\rightarrow GL(n,\mathbb{C})$ be a representation of $G$ and let $χ$ be the corresponding character. Then $\forall g\in G$ we have: If $0<\frac{|χ(...
Fotis's user avatar
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1 vote
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What is known about the minimal absolute value in $\mathbb{Z}[\zeta_n]\setminus\{0\}$?

Here $\mathbb{Z}[\zeta_n]$ is the ring of integers of the cyclotomic field $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is the $n$th root of unity. In my mind $\mathbb{Z}[\zeta_n]$ looks like a grid in the ...
Dusty Dimensions's user avatar
5 votes
1 answer
130 views

Decomposition of rational primes, why work over ring of algebraic integers?

The proof of Thm. 6.1 in this article https://people.reed.edu/~jerry/361/lectures/iqclassno.pdf actually proves the following generalization of Thm. 6.1 beyond quadratic number fields (copied from the ...
D.R.'s user avatar
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1 vote
2 answers
274 views

Why Does (3) Completely Ramify in $\mathbb{Q}(\omega, \sqrt[3]{2})$?

I'm working through this notes on Algebraic Number Theory. In section 2.6 they claim (3) is completely ramified over the larger field: If we know that the ring of integers of $\mathbb{Q}(\omega, \sqrt[...
Johnald's user avatar
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1 vote
1 answer
140 views

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 ...
Ashtart's user avatar
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5 votes
0 answers
264 views

Is there an integral extension of $\mathbb{Z}$ keeping its prime elements?

Let $R$ be a ring. An extension $S$ of $R$ is called prime-keeping if every element $p$ prime in $R$ is also prime in $S$. Consider the ring $\mathbb{Z}$ and the following two extensions: $\mathbb{Z}[...
Sebastien Palcoux's user avatar
1 vote
1 answer
150 views

For which $d$, the algebraic integers of $K =\Bbb{Q}(\sqrt d)$ is a PID?

I know that the matter is settled for $d<0$ and an open problem for $d>0$ but I am asking about already known values. The below theorems are the motivation for asking this question. Let $d$ be ...
Infinity_hunter's user avatar
1 vote
1 answer
230 views

Can every transcendental number be expressed as the infinite sum of a quotient of two polynomials?

Is it possible to express all transcendental numbers (and more generally all real numbers $\in \mathbb R$) as the sum of an infnite series of the form $$\sum_{n=0}^{∞} \frac{p(n)}{q(n)}$$ where $p(n)$ ...
J. Linne's user avatar
  • 3,042
1 vote
0 answers
72 views

Symmetric Expressions in Roots of Polynomial with Coefficients in Integral Domain

Recently I learned about Pisot-Vijayaraghavan numbers, and proofs around them led me to the fact that the sums of $n$th powers of roots of a monic irreducible integer polynomial are integers. I saw a ...
Thomas Anton's user avatar
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2 votes
1 answer
377 views

Proof of Dedekind-Kummer theorem

I will write the statement and then ask my query about part of the proof. Statement Let $p$ be a rational prime. Let $K=\mathbb{Q}(\theta ) $ be a number field where $\theta $ is an algebraic integer. ...
Anonmath101's user avatar
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0 votes
1 answer
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Proving quotient of $\mathcal{O}_K $ with ideal $I$ is finite.

I need help showing the following : Suppose $K$ is a number field of degree $n$ and $\mathfrak{a} $ is a non-zero ideal of $\mathcal{O}_K $. Then $\mathfrak{a} $ as an additive finitely generated ...
Anonmath101's user avatar
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3 votes
1 answer
118 views

Algebraic number theory - finiteness of quotient

In part of a proof in reading that proves $\mathcal{O}_K / \mathfrak{a} $ is finite for any nonzero ideal $\mathfrak{a} $ of $\mathcal{O}_K$. It says that since $\mathcal{O}_K$ is a finitely generated ...
Anonmath101's user avatar
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1 vote
0 answers
57 views

Algebraic number theory proof question

Let $K$ be a number field of degree $n$ and let $H$ be a finitely generated subgroup of $\mathcal{O}_K^+$ of rank $n$. Then $H$ has a $\mathbb{Z}$-basis $\omega_1,...,\omega_n $. $\Delta (\omega_1,...,...
Anonmath101's user avatar
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