# Tagged Questions

Questions related to the algebraic structure of algebraic integers

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### Relationship between Galois extensions of local fields and Galois extensions of numbers fields

Can someone give me a reference where I can find a proof of the following result : Let $L'/K'$ be a Galois extension of local fields, then there exists a Galois extension $L/K$ of numbers field such ...
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### Why $(\alpha-1)^{-1}\le u^2$ where $u$ is a fundamental unit in $\mathbb{Z}[\alpha]$ and $\alpha=2^{1/3}$?

Given $\alpha = 2^{1/3},$ I want to show that $\beta = (\alpha-1)^{-1}$ is a unit in $\mathbb{Z}[\alpha]$ and is between 1 and $u^2$, where $u$ is a fundamental unit in $\mathbb{Z}[\alpha]$. I see ...
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### What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
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### Kernel of a homomorphism is subgroup of squares

Let $\gamma:(\mathbb{Z}/p^m\mathbb{Z})^*\rightarrow \{1,-1\}$ be defined as $\gamma(a)=(\frac{a}{p})$, the Legendre symbol; $p$ is an odd prime, and $m$ is an integer greater or equal to $1$. I have ...
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### What are some strong algebraic number theory PhD programs? [closed]

I am currently applying for PhD programs in the US. My main interests are number theory and algebra. More specifically, I am interested in algebraic number theory (number fields, Galois groups, ...
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### Local fields : $K$ dense in $K_{\frak p}$?

someone can give me a reference where I can find the following result : Let $K$ be a number field and let $\frak p$ be a place of $K$ then $K$ dense in $K_{\frak p}.$ thanks.
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### Decomposition of a primitive ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
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### On the norm formula $N(IJ) = N(I)N(J)$ in an order of an algebraic number field

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module ...
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### Criterion on whether a given ideal of a quadratic order is regular or not

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of rank ...
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### Decomposition of a primitive regular ideal of a quadratic order

Let $K$ be a quadratic number field. Let $\mathcal{O}_K$ be the ring of algebraic integers in $K$. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because ...
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### Galois extensions of Local fields

Let $L / K$ be a Galois extension of local fields. My question: why $L / K$ is necessarily of finite degree ?? thanks.
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### irreducible polynomial in $Z_p [x]$

Let $p$ be a prime integer. prove that the following are equivalent (a) $p$ is a prime in $Z[i]$ (b) $x^2 +1$ is irreducible in $Z_p [x]$ What I know is that if p is a prime in $Z[i]$, $p$ is ...
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### Find all Integers ($n$) such that $n\neq 6xy\pm x\pm y$

I am interested in proving that there exist an infinite number of positive integers ($n$) which are not of the form $$n=6xy\pm x\pm y$$ for $x,y\in\Bbb Z^+$. [Note: The $\pm$ signs above are ...
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### Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$?

This a step in my notes which I can't seem to understand clearly. Why $\mathbb{Z}[\alpha]/23\mathbb{Z}[\alpha] \cong \mathbb{F}_{23}/(x^3-x-1)$ where $\alpha$ satisfies $\alpha^3=\alpha+1$? I see ...