Questions tagged [algebraic-integers]

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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}[...
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1answer
71 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 ...
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1answer
80 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 $p(n)$ ...
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0answers
15 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 ...
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1answer
87 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. ...
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1answer
58 views

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 ...
3
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1answer
96 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 ...
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55 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,...,...