# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Subfield of a cyclotomic number field where a prime $p$ is inert

I am reading this paper by Adleman,Lenstra on finding Irreducible polynomial over Finite field. Here in Section VI(Proof of correctness of Algo B) I came across this argument: Let $q_i$ be a prime ...
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### A prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ lies above/over $p$ if $\mathfrak{p}\cap \mathbb{Z} = p\mathbb{Z}$ [duplicate]

In the concrete case that $\mathcal{O}_K = \mathbb{Z}[i]$, and $\mathfrak{p} = (1+i)$, how to make sense of $\mathfrak{p}\cap \mathbb{Z}$? I want to know if (1+i) lies above/over 2.
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### Finding the ring of integers of $\Bbb Q(\sqrt[4]{2})$

I know$^{(1)}$ that the ring of integers of $K=\Bbb Q(\sqrt[4]{2})$ is $\Bbb Z[\sqrt[4]{2}]$ and I would like to prove it. A related question is this one, but it doesn't answer mine. I computed ...
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### Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
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### Example of field's normal closure that's not Abelian?

Suppose $K$ is a global field, $L/K$ is a field extension, and $M$ = normal closure of $L$ (over $K$). Is it possible that Gal($M/L$) is not Abelian? In all cases I know, $L$ is formed from $K$ by ...
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### Completion and Algebraic Closure

Suppose we start with a valued field $K$ and we want to find a field extension of $K$ that is algebraically closed and complete. The usual process is: Consider the completion $\hat{K}$ of $K$, then ...
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Usually the ring of adeles is defined for number fields: if $K$ is a number field the ring of adeles of $K$ is: $$\mathbb A_K:=\prod_{v}' K_v \;\;\;\;\;\;\;\;\;\;\;\;\;(\ast)$$ where $v$ ranges ...
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### $\mathbb Z$ basis of the module $\mathbb Z [\zeta]$

Given an $n$-th root of unity $\zeta$, consider the $\mathbb Z$-module $M := \mathbb Z[\zeta]$. Does this module have a special name? Does a basis exist for every $n$? And if so, is there an ...
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### Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find the square root for the Number Field Sieve (reference here page 4 first line). It says that for this method the degree of the extension $K/\mathbf Q$ must ...