Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

3
votes
1answer
58 views

Does there exist a finite set of polynomials which do not have roots over any prime field?

The polynomial $x^2 + 1$ has a root in $Z_p$ if and only if $p \not\equiv 3 \mod 4$, and the polynomial $x^2 + x + 1$ has a root in $Z_p$ if and only if $p \not\equiv 2 \mod 3$. So each of the ...
3
votes
0answers
26 views

Are these proposed rules for the canonical factorization of algebraic integers complete?

In $\mathbb{Z}$, the rules are fairly well established, a few minor quibbles notwithstanding. But in, say, $\mathbb{Z}[\sqrt{7}]$, there are, as far as I can tell, no established rules. What I've seen ...
5
votes
2answers
63 views

Show that $\sqrt{-6}$ is irreducible in $\mathbb{Z}+\mathbb{Z}\sqrt{-6}$

Suppose not. Then there exists $\alpha,\beta\in\mathbb{Z}+\mathbb{Z}\sqrt{-6}$ such that $\sqrt{-6}=\alpha\beta\implies\alpha,\beta$ are not units. I'm not really sure where to go from here. Any ...
2
votes
1answer
50 views

Decomposition group of a prime ideal and root of polynomials

Let $f(x)$ be a monic irreducible polynomial with integer coefficient. Let $K$ be the splitting field of $f$ and $\alpha$ one of its roots. Let $p$ a prime number such that $p$ does not divide ...
1
vote
1answer
40 views

$\sqrt {-6}$ is not prime in $\mathbb{Z}+\mathbb{Z}\sqrt {-6}$

Suppose $\sqrt{-6}|(a+b\sqrt{-6})(c+d\sqrt{-6})$. I need to show that $\sqrt{-6}$ does not divide $(a+b\sqrt{-6})$ and does not divide $(c+d\sqrt{-6})$. I thought you might arrive at some ...
0
votes
1answer
24 views

which algebraic number theory book with answers to selected questions for self-study?

All: Can anyone recommend some easy to follow algebraic number theory books with answers (hints) to selected questions for self-study ? If a have no answers to questions, but if you know if some ...
4
votes
1answer
27 views

Abelian Kummer Extension

A field extension of the form $\mathbb{Q}(\zeta_n, \sqrt[n]{\beta})$ where $\zeta_n$ is a primitive $n$th root of unity and $\beta \in \mathbb{Q}(\zeta)$ is called a Kummer extension. Even though ...
2
votes
2answers
31 views

Orderings of $\mathbb Q[\zeta]$

I want to apply an Theorem, but for that I need to know, how many orderings the totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}[\zeta]$ has. I think possible answers are $1$ or ...
1
vote
0answers
53 views

Any general “formula” solutions for higher order polynomial equation?

We know that fifth (or higher) degree polynomial equation has no general solution formula using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of ...
2
votes
1answer
73 views

Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
3
votes
0answers
31 views

Unramification of Ideals in Pure Cubic fields

I need some explanation for this .Let $K=\mathbb Q{\sqrt[3]{m}} $ be a pure cubic field with non square element $\alpha $ in $K$ such that ideal $(\alpha) $ is an ideal square in K. Let $ ...
1
vote
1answer
20 views

If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers? Also, for $O_K$, would subring of $O_K$ be integrally closed?
1
vote
0answers
29 views

Number fields and algebraic integer

Suppose there is a (algebraic) number field $K$. Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$. The elements of $K$ are the root of some monic polynomial ...
0
votes
1answer
28 views

Integral closure as topological closure

For a commutative ring $A$ you can define the integral closure of $A$ as $$\overline{A}^{\operatorname{int}}:=\lbrace x\in \operatorname{Quot}(A)\mid x\text{ is integral over } A \rbrace.$$ Since this ...
0
votes
1answer
30 views

Proving $n^q$ is algebraic when $n\in \mathbb N$ and $q\in \mathbb Q$.

Proving $n^q$ is algebraic when $n\in \mathbb N$ and $q\in \mathbb Q$. Definition: A number is called algebraic if it is the root of a polynomial: $P(x)=a_nx^n+...+a_1x+a_0$ My idea was to approach ...
3
votes
2answers
106 views

Motivating mathematics(particularly algebraic number theory) through historical problems.

Most mathematical textbooks start a subject by going backwards, historically. They will define the terms that were invented to solve a problem in their polished form and then use these definitions and ...
3
votes
1answer
52 views

Hilbert class field of cubic field

Let $K=\mathbb Q(\sqrt[3]7) $ be a pure cubic field with class number 3. I want know how to compute its Hilbert Class Field. I know that its degree of extension is 3. Thank You in advance.
1
vote
0answers
75 views

Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?

All: Can anyone recommend some Computational Number Theory Books, which include software programs for key steps of the proofs of major theorem ? Some computational number theory books only include ...
1
vote
3answers
61 views

Explain why the determinant of $A$ is the index of the subring?

Let $a$ be an algebraic number, whose minimal polynomial has integral coefficients. Let $K = \Bbb Q(a)$ be an algebraic number field. Let $\mathcal O_K$ be the ring of integers in this algebraic ...
2
votes
0answers
29 views

Proof of Hermite-Minkowski's Theorem

I want to prove the following theorem by Hermite and Minkowski: For any given discriminant there are at most finitely many number fields with this discriminant. A very helpful step is that if ...
3
votes
0answers
35 views

When are the coordinates of the intersection points of plane curves actually algebraic conjugates

Suppose $f(x,y)\in\mathbb{Z}[x,y]$ is an irreducible polynomial defining a plane curve. Say I want to find the intersection of this plane curve with the line defined by g(x,y)=y-ax+b. One way to do ...
1
vote
1answer
44 views

Can one characterise a global field geometrically?

A global field is either a a finite extension of the rationals a finite extensions of $F_q(t)$ Alternatively, the second is the function field of an algebraic curve over a finite field. Is there ...
1
vote
0answers
64 views

Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
1
vote
1answer
36 views

Elements in ring of algebraic integers.

If $K$ is a number field and $\mathfrak{O}_K$ the ring of algebraic integers. Let $\mathfrak{p}$ a prime ideal, for each $\alpha\in{\mathfrak{O}_K}$ if ...
2
votes
0answers
40 views

Diophantine equations which are easier to solve using $\mathbb{Z}[i]$ compared to $\mathbb{Z}$

I wanted to know applications of arithmetic in $\mathbb{Z}[i]$ that helps in some problems of $\mathbb{Z}$. I found a wonderful set of notes by Keith Conrad. Now I want to read more on a similar ...
0
votes
0answers
23 views

Algorithms for finding the ring of integers

In the book's Algebraic Number theory, Ian StewarT, Third edition (page 51-52), has the following propositions: Theorem 2.20: Let $G$ be an additive subgroup of $\mathfrak{O}_K$ of rank equal to the ...
3
votes
2answers
39 views

Polynomial transformation of the roots of another irreducible polynomial.

Suppose I have some monic irreducible polynomial $g(x)$ in $\mathbb{Z}[x]$ with distinct roots $r_1,r_2,\dots,r_n$. Suppose $f(x)$ is some other polynomial, not necessarily irreducible. Is there ...
2
votes
1answer
39 views

class number of pure cubic fields and elliptic curves

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will ...
1
vote
1answer
61 views

Hilbert Class Field for pure cubic fields

I am new to class field theory, I want to study Hilbert class field for pure cubic fields. Which is the good source? Thank you in advance.
3
votes
1answer
86 views

Shtukas?$\mbox{}$

Does there exist an exposition of the significance of shtukas for someone who is mathematically literate but is largelly ignorant of Drinfeld modules? This arises in the work of Peter Scholze among ...
4
votes
2answers
81 views

what is the most easy to read Algebraic Geometry book? [duplicate]

All: what is the most easy to read (most accessible) Algebraic Geometry book ? (If possible, I am looking for an introduction book, maybe for undergraduate, and maybe similar to A Friendly ...
4
votes
1answer
49 views

Why a particular ring of integers is not generated by a single element

It says here in the Sage documentation that the ring of integers in the number field obtained from $$f(x) = x^3 + x^2 - 2x + 8$$ is not generated by a single element. How would one go about showing ...
4
votes
1answer
79 views

What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
1
vote
0answers
36 views

Salvaging an Algorithm which Finds the Discriminant of a Number Field

I am reading the book "Algebraic Number Theory and Fermat's Last Theorem" by I. Stewart and D. Tall (3rd edition) and stumbled over one of the problems: Let $K = \mathbf{Q}(\theta)$ where $\theta \in ...
6
votes
1answer
56 views

probability that two randomly selected integers of an imaginary quadratic field of class number 1 are coprime

Given an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$, where $D$ is a Heegner number (1, 2, 3, 7, 11, 19, 43, 67, 163), what is the probability that two randomly selected elements of that fields' ...
3
votes
2answers
39 views

Smallest value taken by a quadratic polynomial in two variables.

Let $p$ be a degree $2$ polynomial with integer coefficients, say $$p(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F.$$ I would like to find an algorithm which solves the following: Problem 1: Given ...
4
votes
1answer
35 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
7
votes
1answer
53 views

What is the smallest $d$ such that $4$ has more than one distinct factorization in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$?

Or if there is no such $d$, how do I prove it? Obviously there is no point to looking for this in an UFD. I've looked in other rings, and each time I think I found it, I divide one of the factors by ...
0
votes
0answers
16 views

Degree of the separable closure of the residue field of a complete field

I'm trying to prove a corollary on ramified extensions but I don't know if my thoughts are true. Let $L$ be a finite extension of a complete discrete valuation field $F$, let ...
1
vote
0answers
36 views

$p$-divisible group of tori

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the ...
6
votes
3answers
110 views

Analogue of $\zeta(2) = \frac{\pi^2}{6}$ for Dirichlet L-series of $\mathbb{Z}/3\mathbb{Z}$?

Consider the two Dirichlet characters of $\mathbb{Z}/3\mathbb{Z}$. $$ \begin{array}{c|ccr} & 0 & 1 & 2 \\ \hline \chi_1 & 0 & 1 & 1 \\ \chi_2 & 0 & 1 & -1 ...
2
votes
1answer
24 views

Problem with the hyperelliptic equation

Suppose $K$ is an algebraic number field with $ [ K : \mathbb{Q} ] = d $. $X, Y , \alpha_1 , \ldots \alpha_n $ are in $O_K$ , i.e. are integral over $\mathbb{Z} $. Suppose that we have the following ...
2
votes
3answers
31 views

Given all the multiples of a prime number $p \in \mathbb{Z}$, is $p\mathbb{Z}$ an ideal of $\mathbb{Z}$?

So I'm having a little trouble understanding the concept of an ideal. The book gives the "classic example" of $2\mathbb{Z}$, the even integers, saying these form an ideal. Would I be correct in ...
2
votes
0answers
55 views

Are there any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? [closed]

I am new to Algebraic Number Theory. I wonder if there is any fundamental new discoveries in Algebraic Number Theory than in ‘traditional’ number theory ? I want to know, beside ‘generalizing’ or ...
0
votes
0answers
51 views

Which is the best book on Goldbach conjecture research

Is there a book which summarizes the major research results in the past, and current research trends, for the Goldbach conjecture? I know, much progress has been made in Analytic Number theory in ...
3
votes
0answers
40 views

Algebraic integers of the form $2\cos (2\pi r)$ and Kronecker's Theorem

Problem: Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are all real and lie in the interval $[-2,2]$. Prove that each root of $P$ has the form $2\cos (2\pi r)$ for some $r\in ...
2
votes
0answers
146 views

$e=1$ in Theorem 30 from Marcus book “number fields”

Theorem 30 in Marcus book states that, if $p\in\mathbb Z$ is an odd prime and $q$ is a prime $\neq p$, then, fixing $d$ as a divisor of $p-1$ we have that $q$ is a $d$-th power $\operatorname{mod}q$ ...
1
vote
0answers
45 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
0
votes
0answers
34 views

Zeros of a complex function

Consider the function $$f(x)= \sum_{j=1}^n b_j e^{i a_jx},$$ where $a_j,b_j$ are algebraic numbers. Denote $A=\{f(x)| x\in \mathbb{R}_{\geq 0}\}$, i.e., $A=f([0,\infty))$. Does this hold? $ 0\in ...
0
votes
1answer
71 views

Can anyone recommend an easy to read algebraic number theory book?

Can anyone recommend an easy to read algebraic number theory book ? I prefer a book with good examples. (hints or answers to selected questions if possible. Not sure if it is possible for a book of ...