Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

1
vote
0answers
34 views

Lucas's Cyclotomic Formula

There are 2 well-known formulas involving Cyclotomic polynomials, which can be described roughly as writing $\phi_n$ as norms of elements in some quadratic extension. They appear in wikipedia and ...
2
votes
1answer
33 views

Splitting a Set of Consecutive Cubes into Equal Subsets

Is it possible to split the set of 14 consecutive cubes $1^3,2^3,\ldots,14^3$ into two subsets of equal sums? There has to be a more efficient approach than brute force, right? Because with brute ...
1
vote
2answers
37 views

Factorization of polynomial in a complete field

Let $k$ be a finite extension of $\mathbb{Q}$ and $|\cdot|$ an absolute value on it (either Archimedean or not). Let $L$ be the completion of $k$ with respect to this value, and take any irreducible ...
10
votes
5answers
218 views

Which of these two factorizations of $5$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{29})}$ is more valid?

$$5 = (-1) \left( \frac{3 - \sqrt{29}}{2} \right) \left( \frac{3 + \sqrt{29}}{2} \right)$$ or $$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$ ...
2
votes
1answer
46 views

Isomorphism of quotient rings

In a course on algebraic number theory, the lecturer says $$\mathcal{O}_K\cong \mathbb Z\left[\frac{1+\sqrt d}{2}\right] \cong\frac{\mathbb Z[x]}{\left( x^2-x-\frac{d-1}{4} \right)}.$$ This ...
1
vote
2answers
62 views

For which values of $d<0$ , is the subring of quadratic integers of $\mathbb Q[\sqrt{d}]$ is a PID?

The "integers" of quadratic field $\mathbb Q[\sqrt{d}]$ , for a squarefree integer $d$ , forms an integral domain . I know that for $d<0$ , the quadratic integers of the quadratic number fields ...
0
votes
1answer
24 views

Finding an integral basis for a lattice defined in terms of an equation modulo p

Let p be a prime number, u relatively prime to p, and $\Lambda := \lbrace (a, b) \in \mathbb{Z}^2 : b \equiv au$ (mod p)$\rbrace$. How then can I find an integral basis $v_1, v_2$ for $\Lambda$? I ...
1
vote
1answer
40 views

If $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ?

If $a,b \in \mathbb Z$ are such that g.c.d.$(a,b)=1$ and if $a^2+3b^2$ is a cube in $\mathbb Z$ , then are $a+\sqrt{-3}b$ and $a-\sqrt{-3}b$ both cubes in $\mathbb Z[\sqrt{-3}]$ ? I cannot use ...
4
votes
0answers
46 views

Prime ideals contained in the union of almost all prime ideals

I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is ...
3
votes
1answer
30 views

Liouville's and Roth's theorems for complex algebraic numbers

Liouville's theorem says that If $\alpha$ is an irrational number which is the root of a polynomial $p$ of degree $d > 0$ with integer coefficients, then there exists a real number $C > 0$ ...
3
votes
0answers
33 views

Field extension of K with unique factorization?

When solving Diophantine equations, often I pass to a number field $K$ and hope that the algebraic integers $O_K$ have unique factorization. Suppose that $O_K$ is not a UFD. Is it possible that there ...
1
vote
1answer
43 views

Prime ideal in the ring of integers of the number field $\mathbb{Q}(x)$ with $x^{3}=2$

In an exercise of the book Algebraic Theory of Numbers by Samuel, one must show that--in the integer ring $\mathcal{O_k}$ of the number field extension $\mathbb{Q}(x)$ where $x^{3}=2$--the ideal ...
1
vote
2answers
60 views

Sums of Consecutive Cubes (Trouble Interpreting Question)

Show that it is possible to divide the set of the first twelve cubes $\left(1^3,2^3,\ldots,12^3\right)$ into two sets of size six with equal sums. Any suggestions on what techniques should be used to ...
3
votes
3answers
78 views

Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$

I am working with the ideals $\mathfrak{p}=\left<2,1+\sqrt{-5}\right>, \mathfrak{q}=\left<3,1+\sqrt{-5}\right>, \mathfrak{t}=\left<3,1-\sqrt{-5}\right>$ and I am trying to prove that ...
4
votes
2answers
52 views

Ring homomorphism takes discriminant to discriminant

Let $R[x] \xrightarrow{\sigma} S[x]$ be a ring homomorphism where $R,S$ are integral domains of characteristic $0$. Is it true that for any monic polynomial $f(x) \in ...
1
vote
1answer
26 views

If the limit of a sequence of algebraic integers is algebraic, does it need to be an algebraic integer?

Consider a sequence $\{\alpha_n\}$ of algebraic integers and let $\alpha = \lim_{n \to \infty} \alpha_n$, where the limit is taken with respect to the usual absolute value in $\mathbb{C}$, and suppose ...
0
votes
1answer
29 views

Finding powers of prime ideals from its generators and understanding generator notation

I am trying to understand ideal notation with pointed brackets and how to use it. For instance, if I had an ideal $\mathfrak{a}=\left<2,1+\sqrt{-5}\right>$, where $2$ and $1+\sqrt{-5}$ are its ...
0
votes
1answer
31 views

Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb ...
0
votes
1answer
42 views

Writing a Gauss sum as a sum over divisors

Let $\chi$ be a Dirichlet character modulo $q$ induced by a primitive character $\chi^*$ modulo $d$ for some divisor $d$ of $q$. Let $n$ be a positive integer, and consider the generalised Gauss sum ...
0
votes
3answers
26 views

Determinant of a Vandermonde matrix of roots of monic polynomial with integer coefficients

Let $p(x)=\sum_{i=1}^n a_ix^i$ with $a_i$ an integer for all $i$ and $a_n=1$ such that $p(x)$ has only real roots, and let $\lambda_1,\ldots,\lambda_n$ be the $n$ roots of this polynomial. Then the ...
0
votes
0answers
47 views

Elementary solution to the Mordell equation $y^2=x^3+9$?

I've recently been wondering how to solve the equation of mordell for k=9, namely: (y^2=x^3+9). It reduced to solving the Thue equation (|a^2-2b^3|=3).Interestingly, the equation has several ...
1
vote
0answers
47 views

Ramanujan conjecture and Langlands program

In the article http://www.thehindu.com/sci-tech/science/the-legacy-of-srinivasa-ramanujan/article2746988.ece, it was mentioned that "This conjecture, later called Ramanujan's conjecture, came to ...
14
votes
5answers
172 views

Why is quadratic integer ring defined in that way?

Quadratic integer ring $\mathcal{O}$ is defined by \begin{equation} \mathcal{O}=\begin{cases} \mathbb{Z}[\sqrt{D}] & \text{if}\ D=2,3\ \pmod 4\\ ...
4
votes
1answer
65 views

Number of finite extensions of $p$-adic number field of given degree $n$

Let $p$ be a prime number, $\mathbb{Q}_p$ the $p$-adic number field. We fix an algebraic closure $\Omega$ of $\mathbb{Q}_p$. Any algebraic extension of $\mathbb{Q}_p$ is assumed to be a subfield of ...
1
vote
1answer
60 views

Motivation for the definition of the Artin Conductor of a representation

I'm trying to figure out what the Artin Conductor of a representation is using Chapter IV.$2\text-3$ of Serre's Local Fields, and I'm struggling to understand the motivation behind its definition. ...
1
vote
1answer
36 views

proving Fermat's theorem on $p = x^2 + 3y^2$

Here is a modern proof from the notes primes presented by quadratic forms. We are interested in $p = x^2 + 3y^2$ so we would like to have something like: $$ p = (x + y\sqrt{-3})(x - y\sqrt{-3}) = ...
1
vote
1answer
42 views

How many positive solutions are there (Positive 3 tuples)?

I want to find how many positive solutions for the Diophantine equation $4x + 2y + 5z = 100$ I found a particular solution $(x,y,z) = (50,-50,0)$ then I found a general solution (basis) $s(-2,-1,2) + ...
0
votes
1answer
12 views

prime ideal of integral closure on the decomposition iff lies above the prime ideal of the ring

I'm having troubles proving the following proposition. In every reference I read, they mark this proposition as "clear" or "trivial", but I am unable to prove it. Some help? Let $A$ be a Dedekind ...
0
votes
0answers
13 views

Number of isotropic vectors of a hermitian form

Good evening, I have a question about isotropic vectors in hermitian spaces and I hope someone can help me out. Let K be a local non-dyadic field and $\pi$ a prime element (so 2 is a unit). Let $h$ ...
2
votes
1answer
21 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...
0
votes
0answers
27 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
0
votes
1answer
48 views

$(1-\zeta_m)$ is a unit in $\mathbb{Z}[\zeta_m]$ if m contains at least two prime factors

We know that for $m=p^r, 1-\zeta_m$ is a prime.Now suppose that m has at least 2 distinct primes appearing in its prime factorization,we need to show that $1-\zeta_m$ is a unit in its ring of integers ...
5
votes
2answers
89 views

Why is $(3,1+\sqrt{-5})^2=(2-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$

I want to show $(3,1+\sqrt{-5})^2=(2-\sqrt{-5})$ in $\mathbb{Z}[\sqrt{-5}]$. It's easy to see $(2-\sqrt{-5})\subset (3,1+\sqrt{-5})^2$ since $2-\sqrt{-5}=3-(1+\sqrt{-5})$ is in $(3,1+\sqrt{-5})^2$. ...
1
vote
1answer
44 views

P-adic valuation for ideals

Let $A$ be a Dedekind domain and $\mathfrak{a},\mathfrak{b}$ be fractional ideals of $A$. Then we know that $\mathfrak{a}$ and $\mathfrak{b}$ can be decomposed into ...
3
votes
1answer
39 views

Describing the Inertia group of a number field

Let $ K \subseteq L$ be number fields and $\pi$ be a prime ideal of $L$. $G = \operatorname{Gal}\left(L/K\right)$ Let $D = \{\sigma \in G\:|\:\sigma(\pi) = \pi\} $ be the decomposition group for ...
6
votes
3answers
106 views

Showing these prime ideals are principal

Let $K=\mathbb{Q}(\theta)$ be a number field where $\theta$ has minimal polynomial $x^3-9x-6$. I had to factorise the ideals $(2)$ and $(3)$ into prime ideals, for which I got $(2) = ...
5
votes
2answers
61 views

Relation between units in $\mathbb{Q}(\sqrt{13})$ and integral solutions to $x^2 - 13y^2 = \pm 1$

I have shown that in the number ring of $\mathbb{Q}(\sqrt{13})$, the units are precisely $\pm \left(\frac{3+\sqrt{13}}{2} \right)^n$. How can one deduce the integral solutions to the related Pell's ...
4
votes
1answer
58 views

Intermediate fieds for a Cyclotomic Polynomial of order $27$?

I would like to determine the Galois structure of the field $K=\Bbb Q(\zeta_{27})$--the rationals adjoined a primitive $27^{th}$ root of unity. That is to say I would like to determine the ...
2
votes
2answers
64 views

A break in symmetry between Algebraic number fields over Q and otherwise

Most of the theorems in algebraic number theory seem to generalize to arbitrary base fields apart from $\mathbb{Q}$ apart from one. The characteristic of the residue field is equal to positive prime ...
1
vote
0answers
64 views

A question about number theory( ask for directions)

I have a problem in mind and I want to know which research field it belongs to, then I can read something more specific(maybe). The problem is : consider two sequences of integers $s_n,r_n$ then ...
1
vote
1answer
22 views

Existence of Conductor for Cyclotomic Extension (pg 200, Serge Lang A.N.T.)

Let $\zeta$ be a primitive $m$th root of unity, $m \not\equiv 2 \pmod 4$, and $K = \mathbb{Q}(\zeta)$. For $p$ prime and unramified i.e. $(m,p) = 1$, I know that the Artin symbol $(p, K/\mathbb{Q})$ ...
0
votes
3answers
71 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
4
votes
1answer
50 views

Intersection of ring and prime ideal

Give an example of an extension $B/A$ of rings, with $B$ an integral domain and a nonzero prime ideal $\mathfrak{p}$ of B such that $\mathfrak{p} \cap A=(0).$ I don't know where to begin with this.. ...
1
vote
2answers
38 views

Finding the minimal polynomial and its conjugates without a matrix

Let $K=\mathbb Q\left(^3\sqrt{5}\right)$ and $\alpha=a+b\left(^3\sqrt{5}\right)+c\left(^3\sqrt{5}\right)^2$. How do I find the minimal polynomial $f_\alpha$ of $\alpha$ over $\mathbb Q$? I am already ...
2
votes
0answers
28 views

Exercise to determine $\mathcal{O}_K$

Exercise: Let $p$ and $q$ be prime with $p \equiv 1 \bmod{4}$ and $ q \equiv 3 \bmod{4}$. Let $K=\mathbb{Q}(\sqrt{p},\sqrt{q})$, determine $\mathcal{O}_K$. I'm stuck with this exercise. As a ...
0
votes
2answers
52 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
0
votes
0answers
55 views

Finding Norm, Trace and Characteristic Polynomial of field extensions

If $K=\mathbb Q(\beta)$ with $^3\sqrt{5}$ and $\alpha=a+b\beta +c\beta^2$, I want to find $N_{K/\mathbb Q}(\alpha), Tr_{K/\mathbb Q}(\alpha)$ and $\chi_{K/\mathbb Q}$ of $\alpha$ for in two different ...
1
vote
1answer
41 views

$\mathcal{O}_K$ analogous to $\mathbb{Z}$?

The definition of $\mathcal{O}_K$ isn't very well explained or motivated in my textbook. Let $K$ be a field a field with $\mathbb{Q} \subset K \subset \mathbb{C}$. $\mathcal{O}_K$ consists of all ...
2
votes
1answer
30 views

Basis for number fields and rings of integers

The notation of algebraic number theory is frustrating for me. What does the notation $\mathbb{Q}(\alpha)$ mean? Assuming $\alpha \notin \mathbb{Q}$, is it simply $\lbrace a+b\alpha|a,b\in \mathbb{Q} ...
0
votes
0answers
15 views

Under what conditions is a tower of quadratic extensions a UFD, GCD domain, or just an Integral Domain?

I have been studying towers of quadratic extensions to $\mathbb Q$ and have noticed the following: $\mathbb Q[\sqrt 2]$ and $\mathbb Q[\sqrt 2][\sqrt 3]$ are unique factorization domains(UFDs), but ...