# Questions tagged [integer-rings]

In algebraic number theory, the ring of integers of a number field $K$ is the ring of all elements of $K$ which are roots of a monic polynomial with rational integer coefficients.

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### Why is $\mathcal{O}_K$ the ring to be considered for factorizing integers?

For $K = \mathbb{Q}[\alpha]$ (with $\alpha$ algebraic over $\mathbb{Q}$), I understand that $\mathbb{Z}[\alpha]$ may be too coarse, and that $\mathcal{O}_K$ (the algebraic integers of $K$) allows more ...
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### Is a ring of integers necessarily Noetherian?

Let $K$ be an algebraic extension field (not necessarily finite) of $\mathbb{Q}$. Let $\mathscr{O}_K$ be the integral closure of $\mathbb{Z}$ in $K$. Then, is $\mathscr{O}_K$ Noetherian? If $K$ is a ...
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### When is the tensor product of rings of integers again a ring of integers?

Given number fields $K$ and $L$, under what conditions does there exist a number field $M$ such that $$\mathcal{O}_K\otimes_{\Bbb{Z}}\mathcal{O}_L\cong\mathcal{O}_M.$$ It is necessary that $K$ and $L$ ...
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### Product of all embbedings of a fractional ideal is principal

Let $F$ be a number field of degree $n$ and $\mathfrak a \subset F$ a fractional ideal. How do you show that $$\prod_{k=1}^n \sigma_k(\mathfrak a)$$ is in the trivial ideal class with a rational ...
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### Why $\mathbb{Z}[\theta]\,/\,\mathcal{P}$ is an algebraic extension over $\mathbb{Z}/p\mathbb{Z}$?

If $f(x)$ is a monic, irreducible polynomial in $\mathbb{Z}[x]$ with $\theta\in\mathbb{C}$ as root, why $\mathbb{Z}[\theta]\,/\,\mathcal{P}$ is an algebraic extension over $\mathbb{Z}/p\mathbb{Z}$? I'...
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