Questions related to the algebraic structure of algebraic integers

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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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Possible dimension of number field

What are the possibilities for the dimension $[E:\mathbb{Q}]$ of a galois extensions $E/\mathbb{Q}$? I think I have a proof that $[E:\mathbb{Q}]$ can be any number. First note that it will be ...
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How to test if a given elliptic curve has complex multiplication

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$ This has ...
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When is one ideal the cube of another?

Let $K=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic fields. Let $ \alpha = a+b\sqrt{-m}$ with $ N(a+b\sqrt{-m})=\beta^3$ (i.e norm of $ \alpha$ is a cube). Then what is the condition to show that ...
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Proof algebraic number fields Janusz, implies that every dedekind Ring is principal?

The book is Algebraic number fields, The ring $R$ is a Dedekind ring and $\mathcal{U}$ is a ideal of $R$. Janusz The first four lemmas is completely clear to me, the While i can't understand the ...
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How to compute the prime ideal factorization of a given ideal in an algebraic ring

I have been working on a problem involving integral ideals in algebraic ring $\mathcal{O}_K$. And it involves the unique factorization of a integral ideal ${I}$ into product of powers of some prime ...
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Prove that if $a$ divides both $2$ and $\sqrt{10}$ in $\mathbb{Z}[\sqrt{10}]$, then $a$ is a unit

Prove that if $a$ divides both $2$ and $\sqrt{10}$ in $\mathbb{Z}[\sqrt{10}]$, then $a$ is a unit. Further, show that you can't express $a$ as $a = 2b + \sqrt{10} c$ where $b, c \in ...
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Prove that if $a +b\sqrt{c}$ is a root of a polynomial in $\mathbb Z$ then $a-b\sqrt{c}$ is also a root of a polynomial in $\mathbb Z$.

Prove that if $a +b\sqrt{c}$ is a root of a polynomial in $\mathbb{Z},$ then $a-b\sqrt{c}$ is also a root of a polynomial in $\mathbb{Z}$. a,b, and c are all integers and c is not a perfect square. I ...
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What is the Euclidean function for $\mathbb{Z}[\sqrt{14}]$?

I've tried a few different pairs of numbers in $\mathbb{Z}[\sqrt{14}]$ and in each case I've been able to find a remainder for which the absolute value of the norm is suitable for the Euclidean ...
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Inertia field of a compositum.

My question is regarding composite field extensions. The problem is motivated but not explicitly stated in the chapter 4 exercises of Marcus' Number Fields, in particular ex. 31 c) We first provide ...
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Struggling to understand algebraic number theory proof

I'm struggling to understand a small part of proof I'm reading on. Let $b_1,...,b_d$ be a basis of the field $K$ over $\mathbb{Z}$ consisting of algebraic integers. $O_K:=\{\alpha\in K|\alpha$ is ...
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Asymptotic for the height of the derivatives of a rational function

Let $\phi=\frac{P(z)}{Q(z)}$ be a homogeneous rational function of degree $d\ge 2$ over $\overline{\mathbb{Q}}$. If $h$ is the absolute logarithmic height, it seems that for each $z\in ...
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2answers
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How to show that if $\alpha \in \mathbb{Z}[\sqrt{2}]$ and $\alpha$ is a unit, then we cannot have $1 < \alpha < 1 + \sqrt{2}$.

How can I show that if $\alpha \in \mathbb{Z}[\sqrt{2}]$ and $\alpha$ is a unit, then we cannot have $1 < \alpha < 1 + \sqrt{2}$. Assuming $\alpha = a + b\sqrt{2}$ is a unit, $1 < \alpha ...
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Algebraic extension of field with characteristic 0

Is it true that any algebraic extension of a field of characteristic 0 also has characteristic 0? Since the field would be infinite, I assume the amount of algebraic integers in it would also be?
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How to calculate $3*5*17*257-3^{16}$ using factorization formulas?

look at this: $2*4*10*82*6562-3^{16}$ It's easy to calculate it with elementary arithmetic. but how to calculate it using factorization formulas?
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Calculating of least significant digit of an expression

I want to calculate te least significant digit (1s place) of following: $ 1+2^{1393} + 3^{1393}+4^{1393} $ How we can calculate this? It's very hard for me!
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Collapsing prime ideals of the form $\langle p, x+ i\rangle$ to a principal ideal

We know $\mathbb{Z}[i]$ is a PID. Also we know that if $p\equiv 1(4)$, $p$ splits in $\mathbb{Z}[i]$ and it factors into a product of prime ideals $\langle p, x+i \rangle \langle p, x-i \rangle$ where ...
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1answer
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Discriminant of a Polynomial over a Local Field

I am trying to prove the local Kronecker-Weber theorem for tamely ramified abelian extensions $L|\mathbb{Q}_p$. At some point in the proof I need to show that $\mathbb{Q}_p(u^{1/e})$ is unramified ...
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0answers
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Surjectivity of the derivation map on Washnitzer Algebra

Let $K$ be a non-archimedean field of characteristic zero and $||.||:K\to \mathbb{R}_{\geq 0}$ be its absolute value. Define the Washnitzer Algebra as: $$W_n=\{\sum_{u\in \mathbb{Z}_{\geq 0}^n} \in ...
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1answer
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Number of places over a number field

Let $\mathbb{K}$ a finite extension of $\mathbb{Q}$ and $\mathcal{O}_\mathbb{K}$ its ring of integers. Assume $\mathcal{O}_\mathbb{K}=\mathbb{Z}[\alpha]$, that is generated as a ring by a single ...
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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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Expressing a number field in a simpler way

Express $\mathbb{Q}(\sqrt{3},\sqrt[3]{5})$ in the form $\mathbb{Q}(\theta)$. Hint: Let $f,g$ be the minimal polynomials of $\sqrt{3}$ and $\sqrt[3]{5}$, respectively. Factor $f$ and $g$ in ...
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$\mathbb Z[\sqrt{-5}]$ is not a UFD [duplicate]

Prove that the ring of integers of $\mathbb Q (\sqrt{-5})$ does not have unique factorisation. Since $-5\equiv 3\pmod 4$, I know that the ring of integers of $\mathbb Q (\sqrt{-5})$ is $\mathbb Z ...
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1answer
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Defining the Artin Map on the Ideles

Let $L/K$ be abelian. There is a natural way to define the Artin reciprocity map on the ideles using the notion of an admissible cycle. I don't want to go into the details of what that is right now, ...
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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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1answer
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Field notation and degree of extension

Consider a field $\mathbb Q (\sqrt5, \sqrt7,\sqrt{35})$ as an extension over $\mathbb Q$. What is the degree of the extension? I am confused by the notation here. Does $\mathbb Q (\sqrt5, ...
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2answers
63 views

The mod $p$ Galois representation of the Frey curve is unramified away from $2, p$

Given a hypothetical solution to Fermat's last theorem for $p \ge 5$ $$a^p + b^p + c^p = 0$$with $a \equiv -1 \pmod 4$, $b$ even, we can write down the Frey Curve$$E: y^2 = x(x-a^p)(x+b^p)$$which has ...
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Norm and Trace of an element is an integer, then element is an integral?

Let $L/K$ be a finite field extension, and let $\{b_1,b_2,...,b_d\}$ be a basis for $L/K$ My notes define $O_k:=\mathbb{B}\cap K$, where $\mathbb{B}:=\{\alpha$ is algebraic|min poly of $\alpha$ ...
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1answer
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Problem solving Neukirch's Algebraic Number Theory, Exercise 1.7.4

I have tried to solve exercise 1.7.4 in Neukirch's Algebraic Number Theory which states that $1+\zeta $ is a fundamental unit of $\mathbb Z [\zeta]$ when $\zeta$ is a primitive $5$th root of unity. I ...
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3answers
260 views

What is wrong with this proof on ring of integers being finitely generated

I have seen some proofs about the theorem that the ring of integers is a finitely-generated $\mathbb{Z}$-module, but I thought I came up with a more straightforward proof. However, I believe there is ...
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1answer
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Can a non-UFD quadratic integer ring have some irreducible numbers that are actually prime?

And if so, is there an efficient way to identify such primes? For example, in $\mathbb{Z}[\sqrt{-5}]$, it's clear that $17$ is irreducible. Among its first fifty multiples ($34$ through $850$) I was ...
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1answer
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When is $\mathbb{Q}[\alpha] \cap \mathbb{R}$ equal to $\mathbb{Q}$?

I'm interested in necessary and sufficient conditions for a nonreal algebraic integer $\alpha$ to satisfy the equality above. I know that if $\alpha$ has prime degree then the equality holds, but I ...
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Introduction to algebraic number theory

How do you get started in algebraic number theory? I am an undergraduate going through a Bachelor in Applied Computer Science and I just finished my first course in Algebra. My university ...
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Inversion of fractional ideals with respect to localization

Let $R$ be a integral domain with field of fractions $K$, $S$ is any multiplicative set in R and $\mathfrak M$ is a fractional ideal of $R$. $$\mathfrak M^{-1}=\{x\in K:x\mathfrak M\subseteq{R}\}$$ ...
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Field Completions

Let $(K,v)$ be a number field with an absolute value. Denote $K_v$ to be a completion of $K$ and $\overline{K_v}$ to be an algebraic closure of $K_v$. Let $E$ be a finite extension of $K$. Every ...
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Algebraic proof of 2nd inequality of Global class field.

s there any 'explicit' proof, preferably without the use of cohomology, of second inequality of global field? By second inequality I mean $[C_K:N_{L/K}C_L]\leq [L:K]$ where $C_K$ is idele class group ...
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Unique factorisation in ring generated by square roots of prime numbers?

Let $\langle\sqrt{\mathbb{N}}\rangle=\mathbb{Z}[\sqrt2,\sqrt3,\sqrt5,\ldots]$ denote the ring generated by the square roots of all prime numbers. Is it it known whether ...
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Some questions about number fields

As a new beginner in algebraic number theory, I am confused with some properties of number fields. First comes some conventional notations. For any number field $K$, let $\mathcal{O}_K$ denote the ...
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quartic subfield of cyclotomic field

I want to know which is the quartic subfield of the cyclotomic field $\mathbb{Q}(\zeta_p)$ where $p$ is an odd prime? it is $\mathbb{Q}(\sqrt{\varepsilon\sqrt{p}})$ where $\varepsilon$ is the ...
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$\mathbb{Z}[\zeta_n]$ 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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Ideal as kernel of a homomorphism in Gaußian integers

Consider the ring $\mathbb{Z}[i]$ of Gaußian integers. The principal ideal $(1+i)$ is maximal ideal in this ring. Since ideals are kernels of some homomorphisms, I would like to see a homomorphism ...
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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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Does $20$ really have three distinct factorizations in $\mathcal{O}_{\mathbb{Q}(\sqrt{-31})}$?

I'm still trying to wrap my mind around the concept of $\sqrt{-1}$, though I think I've gotten to the point where my doubt is more metaphysical than mathematical. I've read a few bits of algebraic ...
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Definitions of valuations in terms of totally ordered group

Wikipedia gives a definition of valuations involving abelian totally ordered groups. So far I have only seen valuations taking values in the real numbers. Is there a reason for this generalization?
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Determine all units in $\mathbb{Z}[\omega] := \{a+b\omega\mid a,b\in\mathbf{Z}\}$ where $\omega = \frac{-1 + i \sqrt{3}}{2}$

My attempt: $N(a + b\omega) = (a + b \omega)(a - b \omega) = a^2 + \omega^2 b^2$ I'm stuck here. Is my approach correct?
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Number of solutions of the congruence, $x-y \equiv z \pmod{n}$, where $x,y$ in a set contain less than $n$ and relatively prime to $n$?

I know number of solutions of the congruence, $$x+y \equiv z \pmod{n},\tag{1}$$ $x,y\in U_{n}$, is $$N(z)=n\prod_{p\mid n}\left(1-\frac{\varepsilon(p)}{p}\right),$$ where $\varepsilon(p)$ = $\left\{ ...
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1answer
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pth root of unity in $p$-adic field

It is well known that $\mathbb{Q}_p(\mu_n)$ is a totally ramified extension of degree $(p-1)p^n$ if $\mu_n$ is a primitive $p^n$th root of unity. However how true is this statement for a finite ...
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1answer
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Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p ...
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There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p$ is a prime and $m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m ...
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Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between ...