Questions related to the algebraic structure of algebraic integers

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2
votes
1answer
236 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
267 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
182 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
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 ...
13
votes
2answers
740 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
338 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
293 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
542 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
236 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
172 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
220 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
651 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
862 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) ...
7
votes
2answers
583 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
144 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
394 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
83 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
168 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, ...
4
votes
2answers
217 views

Split prime in $\mathbb{Z}[\sqrt{14}]$

I have this assertion: if $p$ is a prime such that $p\equiv 11 \pmod{56}$, then $p$ splits in $\mathbb{Z}[\sqrt{14}]$ (the discriminant of $\mathbb{Z}[\sqrt{14}]$ is $56$.) Why? Does $p\equiv ...
13
votes
2answers
548 views

Consequences of the Langlands program

I have been reading the book Fearless Symmetry by Ash and Gross.It talks about Langlands program, which it says is the conjecture that there is a correspondence between any Galois representation ...
1
vote
1answer
109 views

how many embeddings into $\overline{\mathbf{Q}}$ does a given number field have

Fix an algebraic closure $\overline{\mathbf{Q}}$ of the rational numbers. Let $\mathbf{Q}\subset K$ be a number field. I know that the degree $[K:\mathbf{Q} ]$ equals the number of embeddings of ...
18
votes
2answers
1k views

Why do we use this definition of “algebraic integer”?

A number is an "algebraic integer" if it is the root to a monic polynomial with integer coefficients. Artin says (Algebra, p. 411): The concept of algebraic integer was one of the most important ...
2
votes
1answer
214 views

Does product of Galois groups equal to the Galois group of corresponding fields intersection?

Let $k$ be a field, $\bar{k}/k$ be a Galois extension with $G=Gal(\bar{k}/k)$ be an Abelian group(may be infinite). If $K,L$ are intermediate fields, denote $G_K=Gal(\bar{k}/K), G_L=Gal(\bar{k}/L)$. ...
3
votes
1answer
110 views

Totally ramified cyclic extensions of degree $p^a$ of $\mathbb{Q}_p$

It's quite easy to show that the totally ramified extension $\mathbb{Q}_p(\zeta_{p^{a+1}})/\mathbb{Q}_p$ contains a unique subextension $E$ s.t. $E/\mathbb{Q}_p$ is a cyclic extension of degree $p^a$ ...
1
vote
1answer
64 views

Subgroups of groups of order $2^{a-1}$

The context here is the following exercise Let $m=2^a$ with $a > 2$. Show that $\mathbb{Q}(\theta_m)$ contains exactly three quadratic subfields. By Galois theory, this reduces to the problem ...
7
votes
4answers
395 views

Is there a procedure to determine whether a given number is a root of unity?

Let $z$ be an algebraic number of modulus one. Is there a finite procedure that tells us whether $z$ is a root of unity? EDIT: As TonyK and David asked, what I had in my mind is $z$ such that I have ...
-5
votes
2answers
2k views

Is the Birch and Swinnerton-Dyer conjecture solved?

I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some ...
1
vote
1answer
193 views

Tamagawa number conjecture

I heard somewhere that the above formulation of conjecture is for predicting the exact leading term of a L-function at an integer. But i didnt find any reference about how it is stated, anyone please ...
8
votes
2answers
785 views

About cyclotomic extensions of $p$-adic fields

I've been working on the problem of finding the maximal abelian extension of $\mathbb{Q}_5$ that is killed by $5$. In other words, find the abelian extension of $\mathbb{Q}_5$ with Galois group ...
3
votes
1answer
206 views

Class field theory and writing down explicit fields

I'm taking a class in CFT and I'm trying to figure out what the theorems say and what they can be used for to get a "feel" for them. More explicitly, say I take $\mathbb{Q}_p$, so we have the local ...
4
votes
1answer
508 views

rational angles with sines expressible with radicals

An angle x is rational when measured in degrees. sin(x) is can be written using radicals. What are the conditions on x? If nested square roots are allowed? What I know so far: If sin(x) can be ...
15
votes
4answers
2k views

Beginner's text for Algebraic Number Theory

What's good book for learning Algebraic Number Theory with minimum prerequisites? Assume that the reader has done an basic abstract algebra course.
5
votes
1answer
627 views

Why is the determinant equal to the index?

Let $A \subset B$ be integral domains and assume $B$ is a free $A$-module of rank $m$. Define the discriminant of $m$ elements $b_1,\dots,b_m\in B$ as ...
4
votes
2answers
804 views

Roots of unity in a local field

The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as $K=\langle \pi\rangle\times \mu_{q-1}\times ...
2
votes
1answer
223 views

If a prime with prime norm is a split prime, in the number ring PID

If a prime with prime norm is a split prime , in an number ring PID? Example: $5-\sqrt{14}$ in $\mathbb{Z}[\sqrt{14}]$ has norm $11$, it is a split prime in $\mathbb{Z}[\sqrt{14}]$? Why? Thanks
1
vote
1answer
255 views

Fixing Hasse principle

As everyone know that Hasse principle (I am referring to Hasse Local-Global Principle) doesn't work for cubics, but today my question is concerned about: Is there any method or any known theorem, ...
4
votes
2answers
439 views

Is the algebraic norm of an euclidean integer ring is also an euclidean domain norm?

Let K be a finite extension of $\mathbb{Q}$ (a number field) and $\mathcal{O}_K$ its ring of integers. One defines the norm of an element $\alpha\in K$ to be the determinant of the transformation ...
4
votes
1answer
606 views

Intuition and Stumbling blocks in proving the finiteness of WC group

After reading many articles about the Tate-Shafarevich Group ,i understood that "in naive perspective the group is nothing but the measure of the failure of Hasse principle, and coming to its ...
2
votes
1answer
124 views

Double root mod p implies transposition in galois group?

Let $K$ be the splitting field of some irreducible polynomial $f(x)$ in $\mathbb{Z}[x]$, and let $B$ be the integral closure of $\mathbb{Z}$ in $K$. Suppose that, for some prime $p$: $f(x) \equiv ...