Questions related to the algebraic structure of algebraic integers

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6
votes
1answer
325 views

Primes that ramify in a field

Consider the number field $L/\mathbb{Q}$. I know that the only primes $p$ that ramify over $L$ are the ones that divide $\Delta_{L}$, the discriminant of $L$. But what if I can't compute $\Delta_{L}$? ...
3
votes
2answers
749 views

Decomposition and Inertia groups and Frobenius automorphism computations

Let $K = \mathbb{Q}[\sqrt{5}, \sqrt{-1}]$. Calculate the Frobenius automorphisms $\left(\frac{K/\mathbb{Q}}{p}\right)$ for $p$ prime distinct from $2$ and $5$ (which are the only primes that ramify in ...
0
votes
1answer
401 views

Resurrection of my Tamagawa numbers Question, to understand the Formulation of BSD

My previous question was closed very badly for asking the broad and deep things, so I now understand the consequences of asking such questions, so I refrain from asking such questions, so this is not ...
2
votes
3answers
252 views

Kronecker-Weber Theorem

I'm stuck with an article "A simple proof of Kronecker-Weber Theorem" on this website. On page 7, the author proofs that $\mathbb{Q}_p((-p)^{\frac{1}{p-1}}) = \mathbb{Q}_p(\zeta_p)$. While I ...
29
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
3
votes
0answers
79 views

Determining the number of classes

How do can I determine all classes of ideals of $\mathbb{Z}[\sqrt{-104}]$? Or $\mathbb{Z}[\sqrt{-132}]$? (so a list of representatives and showing they are not equivalent, and and that we get all ...
2
votes
1answer
358 views

about the fractional ideal of a field of fractions

In the wikipedia article http://en.wikipedia.org/wiki/Fractional_ideal we read Let $R $ be an integral domain, and let $K$ be its field of fractions. A fractional ideal of $R$ is an $R$-submodule ...
3
votes
1answer
122 views

Characteristic of residue field

Let $\mathcal{O}=\mathbb{Z}[\omega]$ be the ring of algebraic integers in $\mathbb{Q}(\omega)$. It can be shown that $\mathcal{O}$ has a maximal ideal $\mathfrak{m}$ generated by $1-\omega$ (see my ...
2
votes
1answer
544 views

Extending the p-adic valuation

Given a prime $p$, the $p$-adic valuation on the field $\mathbb{Q}$ is the map $\nu:\mathbb{Q}^*\to\mathbb{Z}$ given by $\nu(p^ka/b)=k$, where $a,b$ are prime to $p$. I want to consider extensions ...
7
votes
2answers
874 views

Ideal class group of $\mathbb{Q}(\sqrt{-103})$

I want to calculate ideal class group of $\mathbb{Q}(\sqrt{-103})$. By Minkowsky bound every class has an ideal $I$ such that $N(I) \leq 6$. It is enough to consider prime ideals with the same ...
5
votes
2answers
291 views

Group actions in towers of Galois extensions

Assume we are given an extension of number fields or $\mathfrak{p}$-adic number fields $L/E/K$ where each extension is abelian and $L/K$ is only assumed Galois. Now take any element $\sigma\in ...
8
votes
1answer
2k views

Reading the mind of Prof. John Coates (motive behind his statement)

To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as ...
6
votes
3answers
234 views

When is a local algebra reduced?

Let $k$ be a field and let $A$ be a local $k$-algebra which has finite dimension over $k$. Let $\mathfrak{m}$ be the maximal ideal of $A$ and let $k' = A / \mathfrak{m}$ be the residue field. For ...
0
votes
1answer
62 views

Generating Same Ideal Class

If I have two prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ with $\mathfrak{p} = (a)\mathfrak{q}$ where $(a)$ is a principal fractional ideal (that is, we can have $a$ not necessarily in our ring of ...
6
votes
1answer
779 views

Class number computation (cyclotomic field)

How does one prove that the class number of $\mathbb{Q}(\zeta_{23})$ is divisible by $3$? And afterwards how do you show that it is precisely $3$. Any help? Thanks in advance! //Ok, so I proved the ...
0
votes
1answer
178 views

A reference and an explanation needed?

In my previous question I was asking for a method to construct a global point if we have local points with us which is here, but I got an answer, it didn't serve the entire purpose, but later on due ...
4
votes
1answer
188 views

On Selmer's polynomial

Does this hold? Let $p$ be an odd prime. If $\alpha$ is a root of $x^{p}-x-1$, prove that the ring of integers of $\mathbb{Q}[\alpha]$ is precisely $\mathbb{Z}[\alpha]$ and that this is a PID. ...
2
votes
1answer
139 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
4
votes
1answer
1k views

On determining the ring of integers of a cubic number field

I have the following question: Let $\alpha$ be a root of the polynomial $f(x) = x^3-x+1$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$. As I understand it, I ...
5
votes
1answer
1k views

Integers in biquadratic extensions

Where can I find information (at least examples) about factorization of prime ideals in biquadratic extensions of $\mathbb{Q}$. Right now I have no idea how, for example, find factorization of $(2)$ ...
1
vote
1answer
206 views

A Generalization of Cantor's counting theory

This question may be silly to experts, but I am waiting for a response sir. My question is " Is there any existence of generalized Cantor's counting principle ( i.e the theory that decide ...
2
votes
1answer
227 views

Normalised absolute values on $p$-adic extensions

I have the following problem: show that if $L/K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$ with normalised absolute values $|\cdot|_K$ and $|\cdot|_L$, with $n=[L:K]$, then ...
3
votes
2answers
191 views

Splitting of primes in Galois extensions

I have the following problem: suppose $F/K$ is an abelian (Galois) extension of number fields, Galois group G, and $\mathfrak{p}$ is a prime of K, $\mathfrak{P}$ a prime of F dividing $\mathfrak{p}$; ...
3
votes
1answer
126 views

Factorization of zeta functions and $L$-functions

I'm rewriting the whole question in a general form, since that's probably easier to answer and it's also easier to spot the actual question. Assume that we have some finite extension $K/F$ of number ...
2
votes
1answer
262 views

Decomposition of Tate-Shafarevich group

We all know that Tate-Shafarevich group is defined as $$Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$$ for an abelian variety $A$ defined over a number field $K$, the non-trivial ...
4
votes
1answer
176 views

Compactness theorems of adeles and ideles

I've been reading about adeles and ideles and many authors like Milne and Lang spend some time discussing compactness results related to them. This seemed to me more like a technical point until I ...
0
votes
1answer
203 views

Integral ideals of norm less than the Minkowski Bound

Consider a number field $K$ and suppose I want to find the class group and class number of $K$. One of the first steps is to compute the the Minkowski bound. Suppose our bound is $B$. In all the ...
12
votes
2answers
718 views

How are the Tate-Shafarevich group and class group supposed to be cognates?

How can one consider the Tate-Shafarevich group and class group of a field to be analogues? I have heard many authors and even many expository papers saying so, class group as far as I know is ...
6
votes
2answers
321 views

Splitting of primes in an $S_3$ extension

Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the ...
3
votes
1answer
97 views

Something silly that I'm misunderstanding about ray classes

I just tried to prove Kronecker-Weber and I know the first step is to show that since any modulus in $\mathbb{Q}$ must divide some modulus of the form $\mathfrak{m}=(n)\infty$, so we just need to show ...
8
votes
2answers
289 views

Good undergraduate level book on Cyclotomic fields

I have Lang's 2 volume set on "Cyclotomic fields", and Washington's "Introduction to Cyclotomic Fields", but I feel I need something more elementary. Maybe I need to read some more on algebraic number ...
3
votes
3answers
528 views

A ''corollary'' to the Chebotarev density theorem

Milne's notes on class field theory has the following corollary to the Chebotarev density theorem: If a polynomial $f(X)\in K[X]$ splits into linear factors modulo $\mathfrak{p}$ for all but ...
7
votes
1answer
224 views

Residue at $s=1$ for $\zeta$-functions

Is there any sort of a bound for the magnitude of this residue? I've been looking at some algebraic number theory problems from Princeton's general exam. One of them is the following: Why does a ...
2
votes
1answer
170 views

Galois groups if intermediate extensions and their intersection with decomposition/inertia groups

This is something I started thinking about based on the answer to one of my previous questions. Assume that we have some tower of finite extensions of number fields $L/F/K$ s.t. $L/K$ is Galois. If ...
2
votes
1answer
210 views

Number of totally ramified extensions of $\mathbb{Q}_p$ of degree $n$

I just read the proof of this theorem that $\mathbb{Q}_p$ has finitely many totally ramified extensions of any degree $n$. The proof uses Krasner's lemma and the compactness of a space which ...
5
votes
3answers
625 views

Splitting of primes in the compositum of fields

If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in ...
4
votes
1answer
117 views

Question about the modulus of a number field

Milne defines the conductor of an abelian extension $L/K$ to be the smallest modulus $\mathfrak{m}$ s.t. the Artin map factors as $$\psi_{L/K}:I_K^{\mathfrak{m}}\to \textrm{Cl}_\mathfrak{m}(K)\to ...
20
votes
1answer
814 views

Brave New Number Theory

I suppose this is an extremely general question, so I apologize, and perhaps it should be deleted. On the other hand it's an awesome question. Is it clear exactly how much (assumedly algebraic) ...
6
votes
2answers
550 views

On the class group of an imaginary quadratic number field

Let $d < 0$ be a square-free integer and let $p_{1},\ldots p_{r}$ be the prime divisors of $d$. Let $K := \mathbb{Q}[\sqrt{d}]$ and consider $P_{i} := (p_{i}, \sqrt{d}) \subset \mathcal{O}_{K}$. ...
1
vote
1answer
249 views

Group of units of finite type - related to the factorization of ideals

Let $K$ be a number field, let $A$ be the ring of integers of $K$, and let $P$ denote the set of maximal ideals of $A$. For $p \in P$ and $x \in K^{\times}$ write $v_{p}$ for the exponent of $p$ in ...
3
votes
0answers
82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
1
vote
1answer
143 views

Splitting of primes in the splitting field of a polynomial

Let $K$ be a number field and $f(X)\in K[X]$. Let $E$ be the splitting field of $K$, so that we know that the set of primes splitting in $E$ has density $1/[E:K]$. Milne uses this as the argument ...
2
votes
1answer
110 views

Simple Answer to Showing $N_{k/\mathbb{Q}}(\alpha)=N((\alpha))$

So I'm trying to show that if we have some number field $k/\mathbb{Q}$ and ring of integers $R_k\subset k$, and an element of $R_k$, say $\alpha$, that the field norm of $\alpha$ is equal to the ...
6
votes
1answer
114 views

The conductor and minimal moduli of abelian extensions

Assume that $L/K$ is a finite abelian extension of global fields and $S$ the set of primes of $K$ ramifying in $L$. Then the conductor $\mathfrak{f}(L/K)$ is the smallest modulus s.t. the Artin map ...
1
vote
1answer
371 views

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.
0
votes
1answer
81 views

On unique factorizations of ideals

Using standard notations, let $K$ be a number field and $S = \left\{p_{1}, ..., p_{n}\right\}$ a finite set of non-zero prime ideals of $K$. Let $a$ be a non-zero fractional ideal of $K$. Prove that ...
1
vote
3answers
82 views

The intersection of $O_K$ with $K^\ast$

Let $K/\mathbf{Q}$ be a number field with ring of integers $O_K$. Is $O_K\cap K^\ast = O_K^\ast$? I can't show that the inverse of an element in $O_K\cap K^\ast$ lies in $O_K^\ast$...
0
votes
1answer
65 views

intersections of powers of primes lying over a prime in a Galois extension

Suppose $A$ is a Dedekind domain with fraction field $K$ and $L/K$ is Galois, let $B$ be the integral closure of $A$ in $L$. Let $P$ be a prime ideal in $A$ and let $P_1,...,P_n$ be prime ideals ...
1
vote
2answers
97 views

Question about topology on $K^\times$ in local CFT

I'm trying to parse a page in Milne's CFT notes. The local reciprocity law gives us isomorphisms $$\phi_{L/K}:K^\times/Nm(L^\times)\to \textrm{Gal}(L/K)$$ for all abelian extensions $L$ of a ...
4
votes
0answers
167 views

Prime ideals in galois extensions

This is with reference to proposition 1 in Robert Ash's notes I don't think the Dedekind assumption is necessary. Explicitly, if $A$ is an integral domain with fraction field $K$ and $L/K$ is galois, ...