Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

4
votes
0answers
237 views

Is an algebraic integer all of whose conjugates have absolute value 1 a root of unity? [duplicate]

Possible Duplicate: Are all algebraic integers with absolute value 1 roots of unity? Let $\alpha$ be an algebraic integer. Suppose that all the roots of its minimal polynomial have absolute ...
1
vote
1answer
132 views

Special units of the cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $k$ be a rational ...
1
vote
1answer
311 views

Discriminant of the quadratic subfield of the cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
2
votes
1answer
128 views

Decomposition of a prime number $p \neq l$ in the quadratic subfield of a cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
1
vote
1answer
145 views

Decomposition of a prime number $p$ in a subfield of a cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
1
vote
1answer
146 views

Decomposition of $l$ in a subfield of a cyclotomic number field of an odd prime order $l$

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
4
votes
1answer
177 views

Explicit determination of unramified valuations of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$, $K$ be its field of fractions. Let $p$ be a prime number. Suppose $p$ ...
2
votes
1answer
97 views

A special property of algebraic integers in a subfield of a cyclotomic number field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
3
votes
1answer
176 views

The residue field of a prime ideal of a cyclotomic number field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $p \ne l$ be a prime ...
0
votes
1answer
73 views

A special type of prime decompositions in a subfield of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
-2
votes
1answer
254 views

Periods of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $K = \mathbb{Q}(\zeta)$. Let $A$ be the ring of algebraic integers in $K$. Let $G$ be the Galois ...
2
votes
2answers
865 views

Decomposition of a prime number in a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $A$ be the ring of algebraic integers in $\mathbb{Q}(\zeta)$. Let $p$ be a prime number such that ...
0
votes
1answer
378 views

Prime divisors of the conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
3
votes
1answer
240 views

Gauss' proof of the irreducibility of a cyclotomic polynomial

Let $l$ be an odd prime number. Let $f(X) = 1 + X + ... + X^{l-1} \in \mathbb{Z}[X]$. Probably Gauss was the first man who proved that $f(X)$ is irreducible. I wonder how he proved it.
2
votes
1answer
237 views

A simple property of the norm of an cyclotomic integer

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field. Let $A$ be the ring of algebraic integers of ...
3
votes
1answer
92 views

How to determine whether a given cyclotomic integer generates a prime ideal or not

Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field. Let $A$ be the ring of algebraic integers in ...
2
votes
1answer
87 views

Normality of a certain order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $p$ be a prime number. Suppose the discriminant of ...
1
vote
2answers
70 views

Normality of a certain localization of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $p$ be a prime number. ...
1
vote
0answers
53 views

Criteria for the normality of a certain localization of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $p$ be a prime number. ...
-1
votes
1answer
644 views

Discriminant and conductor of an order of an algebraic number field

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $K$ be the field of fractions of $A$. Let $B$ be the ring of ...
6
votes
3answers
432 views

On the ring generated by an algebraic integer over the ring of rational integers

Let $f(X) \in \mathbb{Z}[X]$ be a monic irreducible polynomial. Let $\theta$ be a root of $f(X)$. Let $A = \mathbb{Z}[\theta]$. Let $p$ be a prime number. Suppose $p$ does not divide the discriminant ...
3
votes
1answer
107 views

Ramification of primes and congruences modulo prime ideals

I have two questions in algebraic number theory which I have difficulty understanding. I would be grateful if something could help me. Let $K$ be an algebraic number field and let $O_K$ be its ring ...
6
votes
1answer
459 views

Positivity of the norm of an element of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be an $l$-th primitive root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field and $\alpha$ be a non-zero element of ...
2
votes
0answers
118 views

Generalized Hensel lifting in a Dedekind domain

The following theorem is known as generalized Hensel lifting(see here). Can we prove this without using $P$-adic completion? Theorem Let $A$ be a Dedekind domain. Let $P$ be a non-zero prime ideal of ...
2
votes
1answer
158 views

Units in number fields with complex embeddings

Assume that we have an algebraic number field with integers $o$, and with a complex embedding $\iota$. What can be said about the image $\iota( o^\times)$ under $\iota$? Is it discrete? Is ...
0
votes
1answer
64 views

All valuations equal one : unit?

Let $F$ be a global field with integers $o$, and let $x \in F$. Does $|x|_v =1$ for all non-archimedean valuations of $F$ imply that $x \in o^\times$.
1
vote
0answers
83 views

Two questions concerning integral dependence

Proposition 2.4 in Janusz's Algebraic Number Fields states that if $R$ is an integral domain with quotient field $K$, $L/K$ a field extension and $b \in L$ algebraic over $K$ with minimal polynomial ...
2
votes
1answer
562 views

Ramification of primes

Let $K \subset L$ be two fields with ring of integers $\mathcal O_K$ and $\mathcal O_L$. If a prime $p$ is totally ramified in $\mathcal O_K$, is it true that $p$ is also ramified in $\mathcal O_L$?
1
vote
0answers
141 views

Intersection of closed sets in $\mathbb{Q}^2$

My question has connection with this question. Let $k>0$ be an integer without square factor. We consider the ring $\mathbb{Z}[\sqrt{k}]$. Let $N(a+b\sqrt{k}):= |a^2-b^2k|$ for $(a,b) \in ...
11
votes
1answer
1k views

How does one graduate from Hecke Operators to Hecke Correspondences?

I've read (skimmed heartily) basic books on the topic of modular forms. (The last being Silverman's Advanced Topics in the Arithmetic of Elliptic Curves.) I strive for an understanding which is as ...
12
votes
2answers
726 views

A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from “Topics in Matrix Analysis” for matrices of algebraic numbers.

It is a well known fact that if $A,B\in M_{n\times n}(\mathbb C)$ and $AB=BA$, then $e^Ae^B=e^Be^A.$ The converse does not hold. Horn and Johnson give the following example in their Topics in Matrix ...
12
votes
0answers
175 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
0
votes
1answer
611 views

Uniqueness, units of the Eisenstein Integers

Let $\zeta$ be the cube root of 1 given by $\zeta=\frac{-1}{2}+i\frac{\sqrt{3}}{2}$ and let $\mathbb{Z}[\zeta]=\{a+\zeta b: a, b\in \mathbb{Z}\}$, called the "Eisenstein integers". How prove the ...
2
votes
1answer
108 views

Common factors of the ideals $(x - \zeta_p^k)$, $x \in \mathbb Z$, in $\mathbb Z[\zeta_p]$

I'm trying to understand a proof of the following Lemma (regarding Catalan's conjecture): Lemma: Let $x\in \mathbb{Z}$, $2<q\not=p>2$ prime, ...
9
votes
4answers
480 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
1
vote
1answer
278 views

An exact sequence on the ideal class group of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated as an $A$-module. It is well-known ...
0
votes
1answer
103 views

Ramification of local field

Let $K$ be finite field, $E=K((x))$ and $F=K((x^n))$, that is $F$ is field of formal laurent series in $x^n$. I know that $E\cong F$ (because you can consider $x\rightarrow x^n$ ) I want to prove if ...
4
votes
0answers
77 views

Using sage to test for squares in residue fields

Let $K$ be a number field, $x \in \mathcal{O}_K$, and $\mathfrak{p} $ a prime of $K$. I want to find out using sage whether or not the reduction of $x$ modulo $\mathfrak{p}$ is a square in the ...
5
votes
1answer
120 views

Embedding torsion units of an order into torsion units of the reduced order.

Let $A$ be an order, i.e. a commutative ring of which the additive group is isomorphic to $\mathbb{Z}^n$ for a certain non-negative integer $n$. Show that there exists an embedding ...
2
votes
1answer
294 views

Algorithm for computing Smith normal form in an algebraic number field of class number 1

Let $K$ be an algebraic number field of class number 1. Let $\frak{O}$ be the ring of algebraic integers in $K$. Let $A$ be a nonzero $m\times n$ matrix over $\frak{O}$. Since $\frak{O}$ is a PID, $A$ ...
6
votes
1answer
520 views

Is there an efficient algorithm to compute a minimal polynomial for the root of a polynomial with algebraic coefficients?

An algebraic number is defined as a root of a polynomial with rational coefficients. It is known that every algebraic number $\alpha$ has a unique minimal polynomial, the monic polynomial with ...
3
votes
1answer
354 views

Norm homomorphism between ideal class groups

I have been working through Number Fields by D.A Marcus, and I'm stuck and need a hint, the question is in chapter 3 question 16 which goes as follows: Let $K,L$ be number fields and $K \subset L$, ...
1
vote
1answer
141 views

Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$

This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.). Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial \[ X^3 -3X + 1. ...
4
votes
1answer
216 views

Are Different Areas of Number Theory That Different

I was wondering if number theorists are "number theorists," or eventually resolve themselves into one of the various branches - i.e., algebraic, analytic, etc. Also out of curiosity, I was wondering ...
22
votes
1answer
683 views

Connectedness of the spectrum of a tensor product.

Let $A$, $B$ be finite free $\mathbb{Z}$-algebras such that $\operatorname{Spec}(A)$ and $\operatorname{Spec}(B)$ are both connected. Is $\operatorname{Spec}(A\otimes_{\mathbb{Z}} B)$ connected?
1
vote
1answer
308 views

A lemma on the integral closure of a Noetherian domain of dimension 1

I need to prove the following lemma(?) which is motivated by this and this. Lemma Let $A$ be a Noetherian domain of dimension 1. Let $K$ be the field of fractions of $A$. Let $B$ be the integral ...
5
votes
2answers
2k views

Main differences between analytic number theory and algebraic number theory

What are some of the big differences between analytic number theory and algebraic number theory? Well, maybe I saw too much of the similarities between those two subjects, while I don't see too much ...
4
votes
3answers
146 views

Showing that an algebraic number is not a root of a real

While answering this question on mathoverflow, I stumbled across a question that I expect may be easily answered by someone knowing a bit more algebra than me. Let's make it really specific. ...
10
votes
2answers
883 views

Ideal class group of a one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $K$ be its field of fractions. Let $B$ be the integral closure of $A$ in $K$. Suppose $B$ is finitely generated $A$-module. It is well-known that B ...
2
votes
2answers
189 views

Is the ring of integers in a relative algebraic number field faithfully flat over a ground ring?

Let $L$ be a finite extension of an algebraic number field $K$. Let $A$ and $B$ be the rings of integers in $K$ and $L$ respectively. Is $B$ faithfully flat over $A$? What if $L$ is an infinite ...