Questions related to the algebraic structure of algebraic integers

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Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for k=9, namely: (y^2=x^3+9). It reduced to solving the Thue equation (|a^2-2b^3|=3).Interestingly, the equation has several ...
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13 views

Number of isotropic vectors of a hermitian form

Good evening, I have a question about isotropic vectors in hermitian spaces and I hope someone can help me out. Let K be a local non-dyadic field and $\pi$ a prime element (so 2 is a unit). Let $h$ ...
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27 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
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54 views

Finding Norm, Trace and Characteristic Polynomial of field extensions

If $K=\mathbb Q(\beta)$ with $^3\sqrt{5}$ and $\alpha=a+b\beta +c\beta^2$, I want to find $N_{K/\mathbb Q}(\alpha), Tr_{K/\mathbb Q}(\alpha)$ and $\chi_{K/\mathbb Q}$ of $\alpha$ for in two different ...
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Under what conditions is a tower of quadratic extensions a UFD, GCD domain, or just an Integral Domain?

I have been studying towers of quadratic extensions to $\mathbb Q$ and have noticed the following: $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 2][\sqrt 3]$ are unique factorization domains(UFDs), but ...
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28 views

Classic proof that $N_{L/K}(P) = p^f$

Let $L/K$ be a finite extension of an algebraic number field $K$. Let $J$ be an ideal of $L$. Hilbert defined the relative norm $N_{L/K}(J)$ of $J$ as $N_{L/K}(J) = \Pi \sigma(J)$, where $\sigma$ ...
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20 views

Isomorphism between polynomial with indeterminate and integral element

If $x$ is an indeterminate, $A$ is a ring and $\alpha$ is in some ring containing $A$, is $A[x]$ is isomorphic to $A[\alpha]$ if $\alpha$ is integral over $A$? What about if $\alpha$ is transcendental ...
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36 views

$\gcd(I, J) = I + J$ and $\mathrm{lcm}(I, J) = I \cap J$

Following is a statement in Marcus Number Fields which I am not able to prove. Let $R$ be a Dedekind domain and $I, J$ be non-trivial ideals in it. Then by unique factorization of ideals in a ...
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Most general conditions under which composition of binary quadratic forms is a group operation?

One direct way to compose two binary forms $f(x,y) = \langle a,b,c\rangle (=ax^2 + bxy + cy^2)$ and $f'(w,z) = \langle a',b',c'\rangle $ where $gcd(a,a',(b+b')/2) = 1$ and $b \equiv b' \pmod2$ is ...
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Construction of $r$ indepentent units for an order

Let $K$ be an algebraic number field over $\mathbb{Q}$ (ie. $K = \mathbb{Q}(\vartheta)$) and $R$ an order of $K$. In $\mathbb{C}$: $f_\vartheta = \prod_{i=1}^n (X-\varrho^{(i)})$. By convention: ...
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Bound on prime and ramification index in local fields.

Suppose $K/\mathbb{Q}$ is a finite extension (degree $n$ say). Choose a prime $\Lambda$ of $K$ lying above $p$ and suppose $K_{\Lambda}/\mathbb{Q}_p$ has ramification index $e$. How "likely" is it ...
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16 views

Is $K_v \subset L_w$ if $w|v$ for number fields $K$ and $L$?

If $L/K$ is an extension of number fields and $w$ and $v$ are places of $L$ and $K$ respectively such that $w|v$ then is there an embedding $K_v\subset L_w$? This is probably a trivial question to ...
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15 views

Why does module index make sense?

On p.10 of Algebraic Number Theory edited by Cassels and Fröhlich, the concept of module index is introduced. My question is that when $M, N$ are not free, how this concept makes sense? Thanks for ...
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33 views

Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

The book Algebraic number fields, Janusz Please, Could you explain the proof of the part b) a little more? Thank you all.
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50 views

Elliptic curves $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic iff $\Gamma=\lambda\Gamma'.$

Let, $\Gamma, \Gamma'$ be $lattices$ of $\mathbb C$, define $elliptic$ $curves$ by $\mathbb C/\Gamma , \mathbb C/\Gamma'$, then $\mathbb C/\Gamma , \mathbb C/\Gamma'$ are isomorphic ...
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31 views

Number of multiples of a polynomial

Given a positive integer $n$, give an asymptotic estimate of the number of polynomials with integer coefficients $p(x)=a_nx^n+\ldots+a_0$ such that $|a_i|\le \binom{n}{i}$ for each $i=0,1,\ldots,n$ ...
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31 views

$\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory

I want to show that $\mathbb{Z}[\zeta_n]$ is a PID for $n=3,4,5$ using Minkowski theory. I know that if the class group is trivial, then it is a PID. Is this helpful to show the claim or how else can ...
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19 views

Ring of integers in a Artin-Schreier extension

It is well-know( see Goldschmidt book: Algebraic Functions and Projective Curves) that a for $q$ a power of $2$ a quadratic separable extension of $\mathbb F_q(T)$ can be written as: $$K:=\mathbb ...
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Polynomial/ Exponential diophantine equation

I am looking for the reference characterizing all the cases when $$an^2+bn+c=2^m$$ has infinitely many positive integer solutions (m,n). Thanks.
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56 views

Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
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Exercise 1.10 from Silverman “The Arithmetic of Elliptic Curves ”

I am having trouble with Silverman's exercise 1.10(b). The converse of (a) is easy because there is no integer solution to the equation when $p \equiv 3$ mod $4$. However, this method does not work ...
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The intersection of a and b is a superset of the product when a and b are ideals

Let a and b be ideals of a ring A. Define $$ab=\left\{{\sum_{j=1}^{n} a_jb_j|a_j\in a,b_j \in b,n \in \mathbb{N}}\right\}$$ Prove that $ab$ and $a\cap b$ are ideals of A, and that $a\cap b \supseteq ...
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24 views

Prolonging a discrete valuation in Serre's Local Fields?

I am really struggling with the concept of prolonging a valuation. Can someone please explain what 'e(E'/K)' is in the exercise below, what it means for K to be complete under a discrete valuation and ...
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18 views

Reference request for valuations in algebraic number theory

I am going through the book "Primes of the form $x^2$ + n$y^2$" and I understand the background material about algebraic number fields and class numbers(from Ireland and Rosen). However, I do not have ...
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29 views

Showing that $A^*$ is and ideal and that “$*$” is multiplicative.

Let $E/F$ be an extension of algebraic number fields and $\mathcal{O}_E$ and $\mathcal{O}_F$ be the ring of integers. Define $$A^*=\mathcal{O}_EA,$$ where A is an ideal of $\mathcal{O}_F$. Prove that ...
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36 views

Generalize a trick with Dirichlet series to algebraic number theory

I am not able to generalize the following equality involving Dirichlet series : ...
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15 views

Set of squares in quadratic forms of a given discriminant.

For quadratic forms of negative discriminant, the set of squares is the same as the principal genus $H$ (forms whose values in $Z/DZ$ is the same as that of $x^2 + ny^2$ or $x^2 + xy + ny^2$ where ...
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Understanding the proof (via primary decomposition) the “ideal factorization theorem” in Dedekind domains

I am trying to understand the outline of the strategy for proving (via primary decomposition) that every non-zero ideal of a Dedekind domain can be expressed uniquely (up to the order of the factors) ...
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25 views

Bounding height of an algebraic number

Let $\zeta_l$ be a primitive $l^{th}$ root of unity($l\geq 5$) and let $H$ be the height of an algebraic number(see for instance page 230 of The Arithmetic of Elliptic Curves by J. Silverman). I ...
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19 views

Representation groups over Dedekind domains

I am interested on groups defined over $O_K$ the ring of integers of a number field $K$. Given a linear representation $T:Gl_N(O_K)\rightarrow Gl(W)$ with $W$ a free $O_K$-module, What are the main ...
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Is it true that $B=[\beta]$ when $B$ is and ideal of $\mathcal{O}$, $\beta \in B$ and $N(B)=|N(\beta)|$?

Let $B$ be an ideal of $\mathcal{O}$ (Ring of integers), $\beta \in B$ and $N(B)=|N(\beta)|$. Does it follow that $B=[\beta]$? I think that this isn't true but I'm struggling to find a counter ...
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20 views

Fractional ideals of $\mathbb{Q}$ prime to $N$

Let $N \in \mathbb{Z}$. What is meant by a fractional ideal $\mathfrak{p}$ of $\mathbb{Q}$ being prime to $N$? Is it that $gcd(\mathfrak{p},N\mathbb{Z})$ contains $\mathbb{Z}$? Let $I_N$ denote the ...
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1answer
41 views

Help unmasking a disguised principal ideal

I recently saw a question on here about trying to generate a non-principal ideal in a principal ideal domain, with the only answer so far saying that if the ring $R$ is a PID, then $\langle e, f ...
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1answer
44 views

Completion of a number field at a complex embedding

Sorry if this question has been asked before. Let $K$ be a number field of degree $n>1$ and $\sigma:K\hookrightarrow \mathbb C$ a complex (non real) embedding of $K$ in $\mathbb C$ giving the ...
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1answer
32 views

Is it true that an equivalent 'absolute value' is an absolute value?

I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by ...
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73 views

Norm in cyclotomic field

Suppose $p$ is a rational prime and $\zeta=e^{2\pi i/ p}$. Prove that the groupp of non-zero elements of $\mathbb Z_p$ is cyclic, show that there exists a monomorphism $\sigma:\mathbb Q(\zeta)\to ...
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Questions on Heath-Brown's paper “Kummer’s Conjecture for Cubic Gauss Sums”

On page 21 in Heath-Brown's paper "Kummer’s Conjecture for Cubic Gauss Sums" (http://eprints.maths.ox.ac.uk/158/1/kummer.pdf), a formula says $$\sum_{j\in \mathbb{Z}[\omega]}f(j)=\sum_{k\in ...
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111 views

Are these proposed rules for the canonical factorization of algebraic integers complete?

In $\mathbb{Z}$, the rules are fairly well established, a few minor quibbles notwithstanding. But in, say, $\mathbb{Z}[\sqrt{7}]$, there are, as far as I can tell, no established rules. What I've seen ...
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Integral closure as topological closure

For a commutative ring $A$ you can define the integral closure of $A$ as $$\overline{A}^{\operatorname{int}}:=\lbrace x\in \operatorname{Quot}(A)\mid x\text{ is integral over } A \rbrace.$$ Since this ...
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30 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
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1answer
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Summation and product over $k$ with $k$ prime to $n$ sought

I just come to a standstill with the following two formulas. If $$E_n=\lbrace k\mid 1\le k\le n\ \&\ (k,n)=1\rbrace$$ then I hope for a closed formula $f(n)$ for those $$\sum_{E_n}k$$ ...
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Number of prime ideals that contain a non zero ideal

In the proof of proposition (12.3) of Neukirchs Algebraic Number Theory, we use the fact that for a one-dimensional noetherian integral domain, there are only finitely many prime/maximal ideals that ...
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Why is a mapping $H \sigma G_{\mathfrak{P}} \mapsto \sigma\mathfrak{P} \cap L$well-defined bijection?

Assume that $K$ is a number field with a ring of integers $\mathcal{O}_K$. Let $L/K$ be an arbitary separable extension, and embed it into a Galois extension $N/K$ with Galois group $G = ...
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$A\alpha_1+\dots +A\alpha_n$ is a $B$-module

Let $A$ be an integrally closed domain with quotient field $K$ and $L/K$ a finite separable field extension. We denote with $B$ the integral closure of $A$ in $L$. Now let $\alpha_1,\dots,\alpha_n\in ...
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Algebraic closures are henselian?

Let $(K,v)$ be a nonarchimedean valued field and $(\widehat{K},\widehat{v})$ be its completion. Let $o$ and $\widehat{o}$ be the valuation rings of $K$ and $\widehat{K}$. Let $K_v$ be the separable ...
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78 views

A question in Neukirch's ANT book

In Corollary II.5.8, Neukirch Algebraic Number Theory(p142, line 11), why $d=v'_p(p)$ where $v'$ is normalized valuation? EDIT In other word, let $K$ be a finite extension of $Q_p$, I.e. a local ...
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The structure of $O_L$ as an $O_K$ module.

This a a second thought after the question: Is $O_L$ a free $O_K$ module? So, if $L/K$ is a finite number field extension,I know that we can find $\beta_1,\dotsc,\beta_n\in L$ such that ...
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divisibility question involving primes

I have a question concerning the following divisibility problem. For any prime $p$ we define set: $\mathtt{V_{p}}:=\Biggl\{F\in\Phi\Biggl|\begin{cases}p^2\nmid ...
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1answer
53 views

Before we consider the prime decomposition

Let $L/K$ be a number field extension. Let $I$ be a prime ideal of $O_K$. How to prove that $IO_L\neq O_L$? It looks there should be a very fast way to see this, but I don't know how.
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Discriminant remains unchanged when reducing mod p

Let $ \theta $ be an algebraic integer and let $g(t) \in \mathbb Z [ t ] $ be its minimal polynomial over $\mathbb Q$. Let $ \bar g (t) \in \mathbb F_p [ t] $ be the same polynomial with coefficients ...