Questions related to the algebraic structure of algebraic integers

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6
votes
1answer
111 views

Artin Reciprocity $\implies$ Cubic Reciprocity

I'm trying to understand the proof of cubic reciprocity from Artin reciprocity as outlined in this well-known previous math.SE question and the link KCd mentions there. However, there's one final step ...
4
votes
1answer
74 views

Does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$?

For prime numbers $p$ such that $p \equiv 11$, $13$, $17$, $19 \text{ mod }20$, does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\...
5
votes
1answer
55 views

Genus of extension $\mathbb{C}(T)(\sqrt{T^n + 1})$

Let $k = \mathbb{C}$ and $K$ is the extension $\mathbb{C}(T)(\sqrt{T^n + 1})$ of $\mathbb{C}(T)$ with $n \ge 2$ an even integer. I suspect that the genus of $K$ is $(n - 2)/2$, but all attempts at ...
3
votes
1answer
106 views

Intersection between two integral closures equals an algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Prove that $A \cap A'...
2
votes
1answer
45 views

If $x \in \mathbb{Z}[\alpha]$, for $\alpha$ an algebraic integer, is $x^{-1} N(x) \in \mathbb{Z}[\alpha]$ too?

Let $\alpha \in \mathbb{C}$ be an algebraic integer. Assume $\alpha \notin \mathbb{Z}$ to avoid triviality. So the minimal polynomial of $\alpha$ has the form $$m(x) = x^n + a_{n-1} x^{n-1} + \ldots +...
0
votes
1answer
65 views

Why is the kernel of a Galois representation an open subgroup?

Assume that $E$ is a completion of a number field. Then either $E = \mathbb{R}$ or $\mathbb{C}$, or $E$ is a finite extension of $\mathbb{Q}_l$ for a suitable prime number $l$. If $E = \mathbb{R}$ or $\...
1
vote
0answers
42 views

The module $\Omega^1_{\mathcal{O}_E|\mathbf{Z}}$

I have been learning about Kähler differentials recently. If $E$ is an algebraic number field, then it is natural to consider the $\mathcal{O}_E$-module of Kähler differentials $\Omega^1_{\mathcal{O}...
4
votes
2answers
106 views

Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
0
votes
0answers
38 views

Is a p-adic number field and a finite algebraic extension of it ultrametric?

An ultrametric space is a special kind of metric space in which the triangle inequality is replaced with $d(x,z)\leq\max\left\{d(x,y),d(y,z)\right\}$. Is a p-adic number field and a finite algebraic ...
4
votes
1answer
441 views

If more than one prime number satisfies a given congruence, must an infinite number of primes satisfy that congruence?

I understand that this is kind of a broad question, but if no affirmative proof is known, can anyone give a counterexample?
1
vote
1answer
23 views

Basis of neighbourhoods in a profinite group

The Krull topology in a Galois group $G$ of a Galois extension $L/K$ is defined taking $\sigma\:G(L/M)$, where $M/K$ varies through the Galois finite subextensions of $L/K$, as a fundamental system of ...
3
votes
0answers
36 views

Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
3
votes
0answers
45 views

The hypothetical fibre over an infinite prime in $\mathbb{A}^1_{\mathbb{Z}}$

This is just a speculative question I've been wondering about; I hope that others may find it interesting too, but if it's too vague please let me know! We can picture $\mathbb{A}^1_{\mathbb{Z}} = \...
2
votes
1answer
45 views

The torsion subgroup of principal units $U^{(1)}$

$\newcommand{\U}{U^{(1)}}$ $\newcommand{\O}{\mathcal{O}}$ $\newcommand{\p}{\mathfrak{p}}$ $\DeclareMathOperator{\char}{char}$ $\newcommand{\N}{\mathbb{N}}$ I have a question about the torsion ...
2
votes
1answer
34 views

Some combinations of quadratic residues

Suppose $x\in\{a,b\}$ solve $x^2=m\bmod q$ and $y\in\{c,d\}$ solve $y^2=n\bmod q$ then when do all combinations of $xy\bmod q$ have same least non-negative residue? Supposing we have $w\in\{e,f\}$ ...
0
votes
1answer
36 views

A question about fields of fraction

I know that defining a field of fraction, is a way to extend a ring to a field, and also I know that $\mathbb{Q}$ is a field of fractions of $\mathbb{Z}$. Or Guassian rationals are field of fraction ...
1
vote
1answer
64 views

Resolving the tedious cubic

The equation given to me is $$4x^4 + 16x^3 - 17x^2 - 102x -45 = 0$$ I'm asked to find it's resolvent cubic which is not so difficult to find. But the problem is that the question further asks to find ...
2
votes
0answers
37 views

Describing integral ideals [closed]

Suppose I have a field $K=\mathbb{Q}(\sqrt{-d})$. How does one describe it's integral ideals?
1
vote
1answer
107 views

Factorization problem in cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1. Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID. Let p be a rational prime, p ...
2
votes
1answer
60 views

the meaning of “ring” in Serge Lang's “Algebraic Number Theory”

On p. 3 of Serge Lang's "Algebraic Number Theory" where he is attempting to define localization he writes "Let $A$ be a ring... Let $K$ be the quotient field of $A$...". Does this mean that he is ...
1
vote
1answer
23 views

Lattices over Dedekind domains.

This is a question related to local fields, particularly on page 48 of Local fields by Serre. Let $A$ be a Dedekind domain with field of fractions $K$. Let $V$ be a finite dimensional vector space ...
-3
votes
1answer
59 views

Really weird numbers [closed]

I am reading 'Algebraic Number Theory' by Chapman and Hall. They (page 94) describe the two sequences: $-1,-2,-3,-7,-11$ $2,3,5,6,7,11,13,17,19,21,29,33,37,41,55,73$ However the numbers mean ...
3
votes
1answer
76 views

number of prime ideals in algebraic integers

Given an algebraic number field $K/\mathbb Q$, let $R = \mathbb Z_K$ be the ring the algebraic integers of that fields. Is it possible to say how many prime ideals there are in $R$? I suspect we ...
2
votes
1answer
41 views

Uniqueness in a Diophantine system

THIS TURNS OUT TO BE FALSE. I POSTED AN ANSWER WITH SOME EXAMPLES. Suppose we have integers $1 < A < B$ with integer $U>0$ and $$ AB = U^2 + 1. $$ Then $A < U < B.$ If we take $$ C = ...
3
votes
4answers
41 views

Verification of a factorization of ideals, $\langle 2 \rangle$

Still going over Alaca & Williams (I might die before I fully understand that book). In $\mathbb{Z}[\sqrt{-21}]$, the factorization of $\langle 2 \rangle$ is $\langle 2, 1 + \sqrt{-21} \rangle^2$....
4
votes
1answer
63 views

Irrational numbers in a cyclotomic extension $\mathbb{Q}(\zeta_N)$

Let $\zeta_N$ be a primitive root of unity so that $\mathbb{Q}(\zeta_N)$ is an $n$-th cyclotomic extension. I am interested in irrational (real) numbers $\alpha$ in $\mathbb{Q}(\zeta_N)$ such that $\...
3
votes
0answers
87 views

Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
4
votes
2answers
122 views

Ideal $\mathfrak p^i$ is not principal

Let c be a postive squarefree integer. Let $K = \mathbb Q(\sqrt{-c})$. Let $p$ be a prime that splits in $K$ and let $\mathfrak p$ be a prime ideal above $p$. I need to prove the following: Prove ...
2
votes
2answers
75 views

How is the $p$-adic Tate module of a formal group defined?

I am familiar with the definition of the $p$-adic Tate module of an elliptic curve defined over a $p$-adic field $k$ (a finite extension of $\mathbb{Q}_p$). But I have also seen some instances where ...
0
votes
0answers
13 views

Close points with bounded minimal polynomial

Let $F$ be an ordered field, let $F'$ denote its real closure. Let $d$ be a natural number and let $A$ be a bounded subset of $F'$, all of whose elements are of degree $d$ over $F$. Is there a map $f:...
2
votes
0answers
38 views

Absolute values on $\mathbb{R}$ [closed]

Two related questions: a) is there some characterization of all the absolute values on $\mathbb{R}$? (similar to Ostrowski for $\mathbb{Q}$) b) are there non Archimedean absolute values on $\mathbb{...
2
votes
1answer
55 views

Proving primes that remain prime in ring of integers when d is congruent to 1 modulo 4

From Artin: When $d$ is congruent $2$ or $3$ modulo $4$, an integer prime $p$ remains prime in the ring of integers of $\Bbb{Q}[$$\sqrt{d}]$ if the polynomial $x^2-d$ is irreducible modulo $p$. ...
4
votes
1answer
64 views

When $n$ is a power of $2$, is $\mathbb Z[\zeta_n +1/\zeta_n]$ a PID?

$\mathbb Z[\zeta_n +1/\zeta_n]$ where $n$ is a power of $2$ and $n>64$ Then we already know $\mathbb Z[\zeta_n]$ is not a PID However, I don't know $\mathbb Z[\zeta_n +1/\zeta_n]$ is a PID. ...
1
vote
0answers
36 views

Computing Hilbert Class Field.

There are some references for computing the Hilbert Class Fields of quadratic extensions, and of cubic extensions. Q1. Is there a general way to compute the Hilbert Class Fields of cyclotomic ...
2
votes
1answer
67 views

How canonical is Gauss's law of composition of forms

Gauss defined the composition of binary quadratic forms $f$ and $g$ to be another binary quadratic form $F$ such that there exist integral quadratic forms $$ \begin{align} r(x_0,x_1,y_0,y_1) &= ...
3
votes
1answer
68 views

Norm of an Ideal.

Let Let $K = \mathbb{Q}(\sqrt{5})$ and $\alpha = \frac{1+\sqrt{5}}{2}.$ How do we describe the ideal $I$ for ring of integers $\mathcal{O}_K$, given by $I = \{ (2+\alpha) a + (1+ 3 \alpha) b : a, b \...
3
votes
2answers
67 views

Using quadratic reciprocity to motivate higher reciprocity laws?

I'm an undergraduate following Neukirch's Algebraic Number Theory; Please do not assume much more than chapters $1$ and $2$ of this book to answer. The topics covered are: algebraic number fields, ...
2
votes
1answer
53 views

The sum of m-th consecutive integers for primes

I’m looking at the sum of m-th powers of consecutive integers i.e. $Sm(p) = 1^m + 2^m +…+(p-1)^m$ I need to prove: Let $p$ be an odd prime. Prove the following congruences: $$\begin{align} Sm(p^2) ...
10
votes
4answers
216 views

Does the equation $x^2+23y^2=2z^2$ have integer solutions?

I would like to show that the image of the norm map $\text N : \mathbb Z \left[\frac{1 + \sqrt{-23}}{2} \right] \to \mathbb Z$ does not include $2.$ I first thought that the norm map from $\mathbb Q(\...
3
votes
3answers
75 views

Question on a permutation of the roots of a Galois number field, dihedral over the rationals

The polynomial $f(x) = x^6 - 7x^5 + 21x^4 - 41x^3 + 63x^2 - 63x + 27$ defines a Galois extension $H$ of $\mathbb{Q}$. The Galois group of the extension $H/\mathbb{Q}$ is dihedral, and depending on ...
1
vote
0answers
29 views

Discriminant of minimal polynomial of an elementary symmetric function of roots

Through various examples I have noticed the following result which seems to be true. I do not see how to prove it, nor how to find counterexamples. Let $P(X)$ be a monic irreducible polynomial with ...
1
vote
0answers
60 views

Weak Mordell-Weil over number fields

I have a question regarding the Mordell Weil theorem a number field $K$. I read the proof of the Mordell Weil theorem in "rational points on elliptic curves" by Tate and Silverman. They presented a ...
0
votes
0answers
24 views

Analogy between group theoretic number theory proofs and function fields

We know that it is possible to prove some number theoretic congruences by using group theoretic tools like Burnside's Lemma for orbit counting. Many of these congruences also hold for $F[x]$ where $F$ ...
0
votes
0answers
25 views

Analogue of orbit counting formula in $F[x]$

We know that if $G$ is a finite group acting on a finite set $\Omega$, then $$\sum_{g\in G}\chi(g)\equiv 0\pmod{|G|}$$ where $\chi$ is the permutation character. I wonder if is there an analogue of ...
0
votes
0answers
65 views

Existence of class modules for finite groups

I'm reading Cohomology of Number Fields by Neukirch et al. Let $G$ be a finite group. A $G$-module C is a class module if, for all subgroups $H \subset G$: 1) $H^1(H,C)=0$ 2) $H^2(H,C)$ is cyclic of ...
1
vote
1answer
41 views

Trace of a function on an elliptic curve

Let $K/F$ be a Galois extension of number fields with Galois group $G$. Let $E$ be an elliptic curve defined over $F$ and $f \in K(E)^{\times}$ be a function. Define the trace of $f$ to be $Tr_{K/F}(...
0
votes
0answers
20 views

Rational prime with a given factorization

Let $\alpha^3-3\alpha+1=0$ and set $K=\mathbf{Q}(\alpha)$. Find a rational prime $p$ such that $p\mathcal{O}_K=P_1P_2$ where $P_1$ and $P_2$ are distinct prime ideals in $\mathcal{O}_K$, or show ...
1
vote
1answer
58 views

Integral and prime ideal in Dedekind domain

Let $A$ be an Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$, and $B$ the integral closure of $A$ in $L$. If $p$ is a prime ideal of $A$, then $pB$ has a ...
8
votes
4answers
129 views

What are some applications of Chebotarev Density Theorem?

Let $L/K$ be a Galois extension of number fields and let $\mathcal{C}$ be a conjugacy class in $Gal(L/K)$. Let $\mathbb{P}(K)$ be the set of all prime ideals in $K$ and let $\left(\frac{L/K}{\mathfrak{...
1
vote
0answers
44 views

Finding a finite Galois cover trivializing a lisse sheaf

Given a lisse $\mathbb{F}_{\ell^r}$-sheaf on a smooth curve $U$ defined ofer $\bar{\mathbb{F}_p}$, Katz says here, in p. 33, that $\mathcal{F}$ "becomes constant on some connected finite etale galois ...