Questions related to the algebraic structure of algebraic integers

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Prime ideals decomposition

How to prove the decomposition law for prime ideals in finite separable extensions of number fields? (J Neukirch - Algebraic number theory- p47, Proposition 8.3)
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Arakelov class group

In book Neukirch Algebraic Number theory. Chapter III theorem 1.12. If $K$ is a number field then $O_K^*$ is finitely generated and $cl_K(\mathcal O_K)$ is a finite group. Theorem 1.12 say that ...
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Galois invariants of the Tate module of an elliptic curve over a number field

Let $K$ be a number field, $E$ be an elliptic curve over $K$, $l \neq p$ be two different prime numbers and $v$ be a place of $K$ above $l$. I am trying to understand the proof of proposition I.6.7 ...
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What are the restrictions in the ramification behavior of a Galois extension of number fields imposed by the Galois group of the extension?

Studying class field theory, I have come across the following Proposition: Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with ...
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1answer
338 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let $K$ be a field, and let $|\cdot|_{1},\dots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
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Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer

I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result: Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic ...
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Proof of the structure theorem

I'm reading through Ian Stewart's book "Algebraic Number Theory and Fermat's Last Theorem" (3rd edition) and I'm having trouble with a bit of the proof of Theorem 1.16 (page 29). The part I don't ...
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Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
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Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
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In what quadrant should the GCD of two Gaussian integers with both real and imaginary parts be?

Examples: $$\gcd(-9 - 3i, -2 + i)$$ $$\gcd(9 + 3i, -2 - i)$$ $$\gcd(-9 - 3i, -2 - i)$$ $$\gcd(-3 - 9i, -2 + i)$$ $$\gcd(-3 + 9i, -1 + 2i)$$ I've put these through Wolfram Alpha and, for some, ...
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A Guide to the Proof of Fermat's Last Theorem from the Modularity Theorem

A few years ago, a friend of mine told me that he had taken an advanced undergraduate number theory class (so something that assumed only a knowledge of algebra and mathematical maturity) which ended ...
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Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
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Two definitions of ramification groups, why are they equivalent?

Let $L|K$ be a finite galois extension and suppose that $v_k$ is a discrete normalized (non-archimedean) valuation of $K$ with positive residue field characteristic $p$, and that $v_K$ admits a unique ...
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1answer
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Dimension of $\mathbb{Q}$-vector spaces $H^m(X, \mathbb{Q})$.

Assume that you can't compute the cohomology group $H^m(X, \mathbb{Q})$ for$$X = \{(x : y : z : w) \in P^3(\mathbb{C}): xy = zw\}$$but you know Weil conjecture. By using Weil conjecture, give the ...
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1answer
85 views

How to show that the norm of a fractional ideal is well-defined?

Sorry. This might probably be a really easy question, but I am only a beginner in algebraic number theory. So, please bear with me. Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring ...
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Given $d \equiv 5 \pmod {10}$, prove $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ never has unique factorization

With the exception of $d = 5$, which gives $\mathbb{Z}[\phi]$, of course (as was explained to me in another question). I'm not concerned about $d$ negative here, though that might provide a clue I ...
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225 views

Multiples of 4 as sum or difference of 2 squares

Is it true that for $n \in \mathbb{N}$ we can have $4n = x^{2} + y^{2}$ or $4n = x^{2} - y^{2}$ for $x,y \in \mathbb{N} \cup (0)$. I was just working out a proof and this turns out to be true from ...
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Are these quotient modules isomorphic?

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. For a non-zero ideal $\mathfrak{a}$ of $\mathcal{O}_K$ and an element $c \in \mathcal{O}_K \setminus \{0\}$ I wonder ...
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Complete field and field extension.

$(K,u)$ be a pair of the field $K$ and its absolute value $u$, $(K_u, \bar u)$ denotes its completion and the corresponding absolute value. Let $L$ be a field containing $K$, $\pi:K_u\rightarrow ...
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unramified quadratic extension of number field

I try to understand the following statement There are only finitely many quadratic unramified extension of a number field $K$ I know by Kummer theory that such extensions are of the form ...
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Characterization of cosine of rational multiples of $\pi$

Given an algebraic number $x$ such that $-1 \leq x \leq 1$ is there a characterization to figure out whether $\cos^{-1}(x)$ is a rational multiple of $\pi$ or not? One characterization would be that ...
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Arithmetic Derivatives: Arithmetic Logarithmic Derivative Problem

In Calculus, whenever we see a constant and want to take the derivative of it, it always is 0. However in Number Theory, we have something called the arithmetic derivative in which we can ...
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Orbits of the $\text{SL}(n,\mathcal{O}_K)$-action on $\mathbb{P}^{n-1}(K)$ for a number field $K$.

I was reading some notes of Keith Conrad where he proves that the number of orbits of the $\text{SL}(2,\mathcal{O}_K)$-action on $\mathbb{P}^{1}(K)$ for a number field $K$ is precisely the class ...
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Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
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Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
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Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.
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How to “floor” in an imaginary quadratic integer ring?

In his answer to this question What is a concrete example to demonstrate that $\mathcal{O}_{\mathbb{Q}(\sqrt{-19})}$ is NOT a norm-Euclidean domain? Robert Soupe essentially looks up in a map to try ...
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Are the ring of integers of the constructible numbers a Euclidean domain?

I suspect that since Euclid uses the Euclidean Algorithm to perform division on constructible numbers in Elements, the ring of integers of the constructible numbers are a Euclidean Domain, but I have ...
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How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
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Irreducibility of cyclotomic polynomials over number fields

Let $K$ be a number field, i.e., a finite extension of $\mathbb{Q}$. For a positive integer $n$, let $\Phi_n(X)$ denote the $n$-th cyclotomic polynomial. Is it possible to say that there exist at ...
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Arakelov Theory and Arakelov curves

There exists a definition of Arakelov Curve in Arakelov theory? My question is because Neukirch (Algebraic Number Theory, Chapter III) defined Arakelov divisors in the set $X=Spec(\mathcal ...
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1answer
324 views

On orders of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
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1answer
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height of formal group of an elliptic curves

I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$. Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times ...
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A consequence of “every norm is equivalent to the sup-norm” in a finite dimensional normed vector space

So, I am reading the following proposition in Neukirch's Algebraic Number Theory: Proposition. Let $K$ be a complete field with respect to the valuation $|\:\: |$ and let V be an $n$-dimensional ...
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Finding the GCD of two Gaussian integers

How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?
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The units of $\mathbb Z[\sqrt{2}]$

How can I show that the units $u$ of $R=\mathbb Z[\sqrt{2}]$ with $u>1$ are $(1+ \sqrt{2})^{n}$ ? I have proved that the right ones are units because their module is one, and it is said to me ...
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Mordell Equation $y^2 = x^3 - 20$. [closed]

Prove that the only integral solutions to $y^2 = x^3 − 20$ are $(x, y) = (6, \pm14)$.
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1answer
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Solving the Mordell Equation $y^2 = x^3 − 2$; what would be a general strategy?

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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Neukirch: Riemann-Roch Theory

In the book Algebraic Number Theory, Chapter III Riemann Roch Theory, theorem 1.12. The two fundamental facts of algebraic number theory, the finiteness of the class number and Dirichlets unit ...
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Ray class group

Can someone please go through a proof of the fact that the ray class group of a number field is finite? I just can't find a nice readable elementary one on the internet... Thanks in advance.
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1answer
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Splitting an ideal lying in a prime ideal

An obviously naive question question I can't answer. Let $A$ be a Dedekind domain, $\mathfrak{p}$ a prime ideal of $A$ and $I$ a non-zero ideal lying in $\mathfrak{p}$. The ideal $I$ can be splitted ...
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1answer
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How to prove a generalized Gauss sum formula

I read the wikipedia article on quadratic Gauss sum. link First let me write a definition of a generalized Gauss sum. Let $G(a, c)= \sum_{n=0}^{c-1}\exp (\frac{an^2}{c})$, where $a$ and $c$ are ...
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Does every ideal class contains a prime ideal that splits?

Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$. Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that ...
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Adelic topology on the group of ideles

The topology on $\mathbb{A}^\times$ is the subspace topology with respect to $\prod_v \mathbb{Q}_v^\times$ and a basis is given by the sets $$\prod_v\Omega_v$$ with ...
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Completion and algebraic closure commutable

The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of ...
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Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
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1answer
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An abelian number field is either totally real or CM-field

The wikipedia article of totally real number fields says: The totally real number fields play a significant special role in algebraic number theory. An abelian extension of Q is either totally ...
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Which mathematical objects generate the zeroes of $L$-functions?

I've studied analytic and algebraic number theory for years and years, and I encountered a hard question about Riemann zeta function and other kinds of $L$-functions - which might be one of the most ...
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Non-CM totally imaginary number fields

Is there a name for the totally imaginary number fields that are not CM-fields? Any important subclass of number fields with that property, or perhaps a reference where those field are studied in ...
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Prove that $\mathbb{Z}[\omega + \omega^{-1}]$ is a PID for $\omega = e^{2\pi i /13}$

As the title suggests, I'm trying to prove that $\mathbb{Z}[\omega + \omega^{-1}]$ is a PID for $\omega = e^{2\pi i /13}$. If we define $R := \mathbb{Z}[\omega + \omega^{-1}]$, then $\text{disc}(R) = ...