Questions related to the algebraic structure of algebraic integers

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4
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1answer
165 views

Ideal classes prime to ramified ideals

Let $K$ be a cubic extension of the rationals, let $O_K$ be the ring of integers of $K$ and let $d$ be the discriminant of $K$. True or false: For every ideal $A$ of $O_K$, there exists an ideal ...
1
vote
1answer
231 views

Constructing a homomorphism from class group to Sha.

In response to my previous question I got a wonderful answer from Prof.Emerton explaining about the similarities between $Ш$ and class group. In order to add something the comments I got from Mr. B R ...
8
votes
2answers
497 views

Elementary proof of $\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $\gcd(n,m)=1$.

In an answer to another question I used the fact that $\mathbb{Q}(\zeta_m)\subseteq \mathbb{Q}(\zeta_n)$ if and only if $m$ divides $n$ (here $\zeta_n$ stands for a primitive $n$th root of unity, ...
4
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0answers
115 views

Ideal class groups and extension of number fields

Let $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ be schemes and $f: X \to Y$ a morphism. By restricting the structure morphism, we get a morphism of sheaves $\mathcal{O}_Y^* \to f_*\mathcal{O}_X^*$. ...
0
votes
1answer
277 views

Valuation of maximal real subfield of cyclotomic field

I'm stuck on the proof of Theorem 4.14 in Washington - Introduction to Cyclotomic Fields. We take a prime power $n = p^m$ and define $\pi = \zeta_n - 1$, $\zeta_n$ a primitive $n$-th root of unity. ...
10
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2answers
489 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
2
votes
2answers
134 views

Equivalence class of an ideal

Let $K$ denote the number field $\mathbb{Q}(\sqrt{15}).$ According to standard lore, we have that $\mathcal{O}_{K} = \mathbb{Z}[\sqrt{15}]$. Moreover, $2\mathcal{O}_{K} = \langle 2, ...
9
votes
1answer
215 views

Class number of $\mathbb{Q}(\zeta_{11})$

I want to compute the class number of $K=\mathbb{Q}(\zeta_{11})$. The Minkowski bound here is < 59, and looking at the factorisation of primes, we can show that the ideal class group is actually ...
3
votes
1answer
254 views

A result of Hermite

He outlined a proof of the fact that there are only finitely many algebraic number fields with a given discriminant. I wish to know whether this fact is somehow related to the assertion that, for any ...
10
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3answers
1k views

How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
13
votes
1answer
377 views

2-Torsion Group Scheme

Consider the elliptic curve $zy^2 + z^2y = x^3.$ I would like to explictly compute the 2-torsion group scheme, $E[2],$ over $\mathbf{Spec}(\mathbb{Z}_2),$ but I'm having a tough time writing down the ...
2
votes
1answer
426 views

How to show that two ideals $I$ and $J$ of $\mathcal O_K$ are equal if $IR=JR$ for some ideal $R$ of $\mathcal O_K$

Suppose $K$ is a finite extension of $\mathbb Q$ and $\mathcal O_K$ is the set of the elements of $K$ which are integral over $\mathbb Z$. Now let $I$ and $J$ be two ideals of $\mathcal O_K$ and ...
10
votes
2answers
561 views

In what senses are archimedean places infinite?

According to Bjorn Poonen's notes here (§2.6), we should add the archimedean places of a number field $K$ to $\operatorname{Spec} \mathscr{O}_K$ in order to get a good analogy with smooth projective ...
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vote
1answer
422 views

Primes that ramify in a cyclotomic extension

Let $F$ be a number field and consider the cyclotomic extension $E = F(\zeta_{10})$ where $\zeta_{10}$ is a primitive 10th root of unity. Why is it true that the only primes of $F$ that ramify in in ...
4
votes
3answers
301 views

Exercise from Marcus on class groups

I stumbled upon this: Let $p$ be a prime of $K$ (a given number field) and let $m$ be the order of $[p]$ in $Cl(K)$. Suppose $\mathcal{P}|p$ is a prime of $L$ (here, $L/K$ is an extension of $K$). ...
5
votes
1answer
211 views

Intersection of a number field with a cyclotomic field

Let $K$ be a number field and $N$ a positive integer. Prove that if the absolute discriminant of $K$ is coprime to $N$, then $K \cap \mathbb{Q}[\zeta_{N}]=\mathbb{Q}.$ This is something that the ...
2
votes
2answers
251 views

Tamagawa numbers and Genus class numbers

I was reading the paper of Prof.Franz Lemmermeyer titled "Pell-conics" which is here, in that the author writes in page 9 that one can define Tamagawa numbers as $$ c_p = \begin{cases} 2 & \text{ ...
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votes
1answer
405 views

An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”

This question is a bit concerned with the Tate-Shaferevich group, lets start defining $C$ as $$C: X^2- \Delta Y^2=4$$ which are generally called as Pell-conics, so all in this question $K$ refers to ...
2
votes
1answer
132 views

Proof of FLT for cyclotomic integers in Edwards

There is a point I don't understand in : Edward, Fermat's Last Theorem. More precisely in the last paragraph of page 173. Let $\alpha =\exp(2i\pi/p)$, and $u=F(\alpha) \in \mathbb{Z}[\alpha]^\times$ ...
5
votes
1answer
402 views

Universality of Tate-conjectures

We all know that Prof.John Tate proposed a set of conjectures(along with Prof.Emil Artin) formally spread under the name of "Tate conjectures", they have a wide range of influence on various fields of ...
3
votes
1answer
186 views

On L functions and primitive Dirichlet characters

I'm walking through the proof (a proof, better said) of Dirichlet's theorem and I'm having trouble explaining this. I'll state it as an exercise. First, say $K$ is a quadratic extension of ...
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0answers
80 views

2-Brauer characters of the symmetric group $\mathfrak{S}_3$

In a previous question, I asked how to compute Brauer characters of the alternating group $\mathfrak{A}_3$; the answer to this question provided a solution for all cyclic groups. I would now like to ...
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1answer
343 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
815 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 ...
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1answer
413 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 ...
3
votes
3answers
264 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
80 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
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1answer
386 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
124 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
616 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
955 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
303 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
236 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
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1answer
63 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
822 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 ...
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1answer
182 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
190 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
140 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 ...
5
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 ...
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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 $(2)$ ...
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vote
1answer
208 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
241 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
197 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
132 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
271 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
183 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 ...
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1answer
211 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 ...
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votes
2answers
755 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 ...