Questions related to the algebraic structure of algebraic integers

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Ideal as kernel of a homomorphism

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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9 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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2answers
37 views

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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1answer
13 views

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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16 views

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 known number of solution of the congruence, $x+y \equiv z \pmod{n}$,$x,y\in U_{n}$ is $N(z)=n\prod_{ p\backslash n}\left(1-\frac{\varepsilon(p)}{p}\right)$, ...
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1answer
37 views

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
37 views

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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37 views

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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1answer
44 views

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 ...
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32 views

Why do modular curves parametrise elliptic curves?

Let $Y(n)=\Gamma(n)/H$, where $H$ is the upper half plane. In these lecture notes http://math.uga.edu/~pete/modularandshimura.pdf , the author makes the following statement: "$Y(n)$ parametrises ...
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1answer
36 views

Eigenvalue of multiplication in a number field.

Let $\mathbb{K}$ be an algebraic number field. $b \in \mathbb{K}$ defines a linear transformation $\phi(b)$ on $\mathbb{K}/\mathbb{Q}$ by multiplication - $\phi(b)(x):=bx$ for all $x\in\mathbb{K}$. ...
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0answers
60 views

Idele for a rational number $q=\frac{63}{550}$ [on hold]

Wikipedia, in its article "p-adic number", has taken an arbitray number $x= \frac{63}{550}$ to show the p-adic absolute value with respect to different primes. Obviously, the p-adic absolute value is ...
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32 views

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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1answer
17 views

Trouble understandingthat the special set $\mathbb{B}$ has the following properties

Let $\mathbb{B}:=\{\alpha\in\mathbb{C}|$The minimum polynomial of $\alpha$ lies in $\mathbb{Z}[x]\}$ In my notes for Algebraic number theory it's proven that the set $\mathbb{B}$ has the following ...
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1answer
42 views

Two question about ${\mathbb Q}(\alpha)$ for $\alpha$ := $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$

Let $\alpha$ = $\sqrt[n]{7}$ + $\sqrt[n+3]{7}$ with $n$ not divisible by $3$. Prove that $[{\mathbb Q}(\alpha) : {\mathbb Q}] = n(n + 3)$. Conclude that $\alpha$ is constructible if and only if $n = ...
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2answers
40 views

Is there a subset of natural numbers with a special property

Let set $A$ be an infinite big subset of the set $\mathbb{N}$ (set of natural numbers),it is not equal to $\mathbb{N}$ and it has the following property: For every $a$ that is not from the set $A$ ...
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1answer
63 views

Fermat's Last Theorem and the Projective Curve $C_N$

In "Silverman & Tate" on page 230 in the appendix on projective geometry, there is the remark: The $N$th Fermat curve $C_N$ is the projective curve: $$C_N: X^N + Y^N = Z^N$$ ...
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0answers
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Do you know any answer for equation y^2 = x^3 + k? [duplicate]

As you know, the equation y^2 = x^3 + k for k like (4n-1)^3 - 4m^2 that m , n are integers & no prime number that p is congruent to 1 modulo 4 count m, don't have any answer & it's proof is by ...
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1answer
174 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if ...
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1answer
44 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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64 views

Prove that for $n\ge 6$ there is always a solution

We have the eation $\frac{1}{a_1^2} + \frac{1}{a_2^2}+...+\frac{1}{a_n^2}=1$. Prove that the equation has for $n\ge 6$ always natural solutions. Any $\frac{1}{x^2}$ can be displayed as sum of 4 ...
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2answers
45 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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39 views

Solve $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$

Prove that for every natural number $m$ there is a natural solution for the eqation $\frac{x}{m}=\lfloor{x^{2/3}}\rfloor+ \lfloor{x^{1/2}}\rfloor+1$ Beside the typical inequality I can't get nothing ...
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2answers
121 views

Topology on $Z_p$

let $Z_p$ denote the $p$-adic integers, then it has a topology as a subspace of $\prod_nZ/p^nZ$, where $Z/p^nZ$ is given the discrete topology. (reference I posted before: Why $Z_p$ is closed.) Now ...
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0answers
42 views

Leads for penetrating the field of Algebraic number theory

I need to rapidly get up to speed on the following topics, for the purposes of an internship: Global and local fields. Localization. Number fields, function fields, etc. Ring of integers, field of ...
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Why $Z_p$ is closed.

Let $A_n=\mathbb{Z}/p^n\mathbb{Z}$ be a ring and $p$ is prime, $\phi_n: A_n\rightarrow A_{n-1}$ be a natural homomorphism (Elements of $A_{n}$ define in an obvious way elements of $A_{n-1}$). Define ...
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43 views

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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1answer
51 views

Constructing idele from a rational number.

I am a novice to concept of idele ,despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance,suppose I take q=396000. Does it mean that the idele q= ...
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1answer
31 views

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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0answers
19 views

Ideals of the quotient ring of A [duplicate]

Let A be a ring and b be an ideal of A. The quotient ring of A by b, denoted A/b is the ring of all equivalence classes A + b. Prove that the assignment $$c → c/b$$ induces a one-to-one ...
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Definiton of topology in Galois group

In the book Algebraic numbers and algebraic functions of E. Artin, chapter six, page 103-104. Artin never says that $\Omega$ is of characteristic $0$ or $p>0$. 1) But by defining of ...
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3answers
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Prove that $2$, $3$, $1+ \sqrt{-5}$, and $1-\sqrt{-5}$ are irreducible in $\mathbb{Z}[\sqrt{-5}]$.

So the Norm for an element $\alpha = a + b\sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$ is defined as $N(\alpha) = a^2 + 5b^2$ and so i argue by contradiction assume there exists $\alpha$ such that $N(\alpha) ...
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1answer
28 views

algebraic conjugate and sum of roots of unity

In above lemma, why $|a'| \leq 1$ still holds? I didn't see how it relates to "algebraic conjugate of a root of unity is also a root of unity", since $a$ is the sum of unity. (definition of ...
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1answer
96 views

Solutions to $y^2 = x^3 + k$?

As you know, the equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 count m, don't have any answer and its proof can ...
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1answer
151 views

Is $\sum\frac1{p^{1+ 1/p}}$ divergent?

Is $\displaystyle\sum\frac1{p^{1+ 1/p}}$ divergent? How can we prove that it is divergent or convergent in analytic number theory? I know what bound of the n-th prime number is, and that its order is ...
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2answers
83 views

A question about rings of algebraic integers

Let $R$ be a subring of the field of algebraic numbers. If $R\cap \mathbb{Q}= \mathbb{Z}$, does it follow that all of the elements of $R$ are algebraic integers?
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1answer
36 views

Basis for rank $n$ ring containing $1$.

Suppose $L$ is a finite (separable?) extension of a number field $K$, $\mathcal{O}_K$ is the ring of integers of $K$, and $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$. How can ...
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1answer
23 views

Are the discriminant of abelian cubic extensions of $\Bbb Q$ equal to the square of their conductor?

Here the conductor $N$ of an abelian extension $\Bbb Q \subset K$ is the smallest positive integer $N$ such that $K \subset \Bbb Q(\zeta_N)$. Thanks to class field theory there is an equivalence ...
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1answer
64 views

Fundamental Unit In Algebraic Fields

Say we have an algebraic field with an infinite amount of units. If one multiplies two units one obtains another unit. In some cases, all units are powers of just one unit ( that's the fundamental ...
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1answer
59 views

Prove Composite Infinitely Often

Suppose we let $f(x) = ax^2 + bx + c$ be a non-constant polynomial, and assume that $a$, $b$, and $c$ are integers. Prove: There are infinitely many integers $n$ such that $f(n)$ is composite.
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1answer
42 views

Classification of Discrete Subrings of $\mathbb C$

I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using ...
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1answer
31 views

Cover of $\Bbb P^1_k$ ($k$ sep. closed) unramified away from $\infty$ and tamely ramified at $\infty$

I'm reading a paper where the authors claim that for a separably closed field $k$ of characteristic $p>0$, there's no cover of $\Bbb P^1_k$ unramified away from $\infty$ and tamely ramified over ...
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Integration over ideles over $\Bbb{Q}$ , Tate`s thesis special case

Let $f\in S(A_\Bbb{Q})$ that is $f$ is adelic Schwartz-Bruhat function over $\Bbb{Q}$, such that all its components in the finite places are characteristic functions of the corresponding ring of ...
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1answer
85 views

Algebraic number theory exercise

I'm trying to do the following exercise from Neukirch's Algebraic Number Theory (exercise 2, $\S 1$, chapter 1): Show that, in the ring $\mathbb{Z}[i]$, the relation ...
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0answers
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Tate thesis : Global functional equation [closed]

It will be very helpful if someone tells me how to do EXERCISE 1 here. I have done part $2$. I cannot do part $1$ and part $3$. I tried part $1$ by decomposing $Z(f,s)$ as the product of its local ...
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1answer
42 views

Explicitly computing uniformisers of local fields

Consider the field tower $L/K'/K$ where $L=\mathbb{Q_3}(\xi,2^{1/3})$, $K'=\mathbb{Q_3}(\xi) $ and $K=\mathbb{Q_3}$. Here, $\xi$ is a primitive cube root of unity, and $\mathbb{Q_3}$ is the 3-adics. ...
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0answers
60 views

Any good supplementary text on Algebraic Number Theory?

I'm going to study algebraic number theory by myself with the following texts: Algebraic Number Theory by Cassels, Frolich Number Fields by Daniel Marcus But I'm not sure whether or not I should ...
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1answer
38 views

What is a system of representatives of the residue field in its ring R?

Let R be a complete discrete valuation ring, with field of fractions K and residue field $\hat{K}$. Let S be a system of representatives of $\hat{K}$ in R. Can someone please explain to me what a ...
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0answers
19 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 ...