Questions related to the algebraic structure of algebraic integers

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About Local fields

Let $\widehat{L}/\widehat{K}$ be an extension of local field we know there are a number field $L$ and a place $\frak P$ of $L$ such that $\widehat{L}=L_{\frak P}.$ 1) How we can prove that $\widehat{...
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Points on elliptic curves

I am learning elliptic curves theorem and I have read in more papers that for two distinct points $P$ and $Q$ there is always point $R$ such that $P+Q+R = \infty$. I know that this point should be ...
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Why does taking the residue commute with the discriminant if $B$ is free over $A$ and not in general?

Let $K$ be a number field, $A$ its ring of integers, $L/K$ be a finite extension, and $B$ the integral closure of $A$ in $L$. Lemma (residue of the discriminant): Assume $B$ is free over $A$, let $a$ ...
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114 views

Ring of integers of unramified extension

Let $L/K$ be unramified extension of local fields, and $k,l$ - their residue fields, $l=k(\overline \alpha)$. Is it true that $\mathcal O_L=\mathcal O_K[\alpha]$? And can it be proved if it's true.
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$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
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Hensel's lemma in $\mathbb Z_2$

Can you give me a concrete example for a quadratic form $$ f(x,y)=ax^2+bxy+cy^2 \in \mathbb Z_2[x,y] $$ which has a primitive solution $(x^*,y^*) \in \mathbb Z_2 \times \mathbb Z_2$ (mod 4) with the ...
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Norms on $k(T)$ equivalent under automorphism

Let $k$ be a field and $K = k(T)$ the field of rational functions in one variable over $k$. Consider the two norms on $K$, which restricted to $k$ are trivial: $$\begin{aligned}& ||x||_1=s^{-v_P(...
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When exactly is the splitting of a prime given by the factorization of a polynomial?

Let $L/K$ be an extension of number fields with $L=K(\alpha)$, where $\alpha\in \mathcal{O}_L$. Let $f\in \mathcal{O}_K[x]$ be the minimal polynomial of $\alpha$ over $K$. Let $\mathfrak{p}\...
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1answer
55 views

Example of number field $K$ such that $[K(\zeta_m):K] < \phi(m)$

Actually two questions: 1) What is an example of number field $K$ such that $[K(\zeta_m):K] < \phi(m)$? 2) In class we discussed that if $L$ is a field of characteristic zero, and $K$ is the ...
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How to compute the class group of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result ...
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Etymology of 'finite place'

In study of algebraic number theory one often comes across the terms 'infinite' and 'finite' places, referring to the archimedean and non-archimedean valuations of your field, respectively - but I ...
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1answer
109 views

The inertia group for the archimedeans places

If $K/k$ is Galois (not necessarily finite), $w$ is an archimedean place of $K$, and $v$ is the place of $k$ below $w,$ then we define (the inertia group ) : $$T(w/v) = \{\sigma\in Gal(K/k) \hspace{...
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1answer
68 views

set theory, show sets are not of equal cardinality - check my proof

question from exam in set theory: let $M$ be the set of all real numbers x that satisfy: $cx^2+bx+a=0$ where $a,b,c \in Z$ (Meaning they are integers) and $c$ is not $0$. We will define $K = \{sm+t |...
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1answer
166 views

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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1answer
60 views

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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1answer
68 views

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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1answer
80 views

Norm of a regular ideal of an order of an algebraic number field which is Galois over $\mathbb{Q}$

Let $K$ be an algebraic number field of degree $n$. Let $\mathcal{O}_K$ be the ring of algebraic integers. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $...
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1answer
128 views

What is an example of a Dedekind ring that is non-principal?

Prop. 15 of Serge Lang's ANT shows that a sufficient condition for a Dedekind ring $R$ to be principle is that it only have finitely many primes. To give an outline of the argument, one starts with a ...
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Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let $\...
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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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125 views

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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191 views

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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1answer
246 views

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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1answer
176 views

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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1answer
83 views

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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346 views

Norm of the product of two regular ideals of 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. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $...
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4answers
231 views

is the number algebraic?

Is the number $\alpha=1+\sqrt{2}+\sqrt{3}$ algebraic? My first attempt was to try a polynomial for which $p(\alpha)=0$ for some $p(x)=a_{0}+a_{1}b_{1}+\cdots +b_{n-1}x^{n-1}$ i. e $x=1+\sqrt{2}+\sqrt{...
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2answers
142 views

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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Determination of the prime ideals lying over $2$ in a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $x_1,\...
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1answer
146 views

Determination of the prime ideals lying over an odd prime in a quadratic order

We need some notation before we state the problem. Let $K$ be a quadratic number field, $d$ its discriminant. Let $R$ be an order of $K$, i.e. a subring of $K$ which is a free $\mathbb{Z}$-module of ...
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182 views

Finding number of positive integral solutions of $x^4-y^4=3879108$

Find the number of positive integral solutions of $$x^4-y^4=3879108$$ $$3879108=36*277*389$$ I tried simplifying factors of $3879108$ to get terms in the form of $x^4-y^4$. However, I am unable to ...
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Conductor of a quadratic order

We need some definitions to state the problem. Let $B$ be a commutative ring, $A$ its subring. We denote by $(A : B)$ the set $\{x \in B | xB \subset A\}$. $(A : B)$ is an ideal of $B$. It is ...
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65 views

Cyclotomic integers with complex norm $1$.

Let $R=\Bbb{Z}[\zeta_n]$. What are the elements in this ring with complex norm $1$? Are only $\zeta_n^i$ all such elements? What if $n$ is odd prime?
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Condition for $a, b + \omega$ to be the canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. By this question, $1, \omega = \frac{(D + \sqrt D)}{2}$ is a basis of $R$ as a $\mathbb{Z}$-module. Let $I$ be a ...
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1answer
168 views

Canonical basis of an ideal of a quadratic order

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, $D$ its discriminant. I am interested in the ideal theory on $R$ because it is closely related to the theory of binary quadratic forms ...
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2answers
61 views

Isomorphism of $\mathcal{O}_K$-modules

I came across the following claim in K Conrad's notes: Let $L/K$ be a finite extension of number fields, For fractional ideals $\mathfrak{a}, \mathfrak{b}$, and $\mathfrak{c}$ of $\mathcal{O}_L$, with ...
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2answers
288 views

Cyclotomic Integers?

Let $K=\mathbb{Q}(\zeta_{p^\infty})$ be the extension of $\mathbb{Q}$ obtained by adjoining all $p-$power roots of unity. My question is : how to show that the the ring of integes of $K,$ $O_K$ is ...
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Smart way to compute decomposition of a prime

Let $f(x) = X^5 -4x +2$. I am wondering if there is a way to compute the decomposition of $(5)$ in $K = \mathbb{Q}[X]/(f(X))$ without computing the decomposition of $f$ in $\mathbb{F}_5$. What I'm ...
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1answer
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Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers. I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and ...
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Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$?

Is it true that $(4+\sqrt{14})(4-\sqrt{14}) = (4+\sqrt{14})^2$ in $\mathbb{Q}(\sqrt{14})$? I am going through the solution of a problem I'm working on and this seems to be what they are saying. If its ...
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2answers
374 views

If $\alpha$ and $\beta$ are algebraic integers then any solution to $x^2+\alpha x + \beta = 0$ is also an algebraic integer.

An algebraic integer is a complex number that is a root of a monic polynomial with coefficients in $\mathbb{Z}$. Let $\alpha$ and $\beta$ be algebraic integers. Then any solution to $x^2+\alpha x ...
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Find all Integers ($ 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 ...
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2answers
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Ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$

Show that the ideal generated by 3 and $1+\sqrt{-5}$ is not a principal ideal in the ring $\Bbb Z[\sqrt{-5}]$. I fail to understand how can 3 and $1+\sqrt{-5}$ generate an ideal. $$(3,1+\sqrt{-5}) =...
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Examples of idelic Artin map

I do not know of any source on class field theory which explains the idelic Artin map through a couple of examples. Please provide me some.
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Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
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107 views

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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1answer
481 views

Intuition for Class Numbers

So I've been thinking about the analytic class number formula lately, and class numbers in general and I'm trying to develop a good intuition for them. My basic question, which may be too general/...