# Questions tagged [algebraic-integers]

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

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### 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 ...
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### 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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### 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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1 vote
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### 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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### 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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### 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 ...
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### 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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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,...,...