Questions related to the algebraic structure of algebraic integers

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9
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3answers
173 views

Something screwy going on in $\mathbb Z[\sqrt{51}]$

In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out. But when I try the same thing for ...
9
votes
4answers
1k views

(Simple?) applications of Class Field Theory?

Does anyone know any simple/nice applications of class field theory? I would really like to find one related to diophantine equations, but anything you got would be good. Thanks
9
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2answers
288 views

Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$

On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with ...
9
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3answers
106 views

What is $\frac{1}{1+\sqrt[3]{2}}$ in $\mathbb{Q}(\sqrt[3]{2})$?

Since $\mathbb{Q}(\sqrt[3]{2})$ is a field, any number $\neq 0$ has a reciprocal. How then to write $\frac{1}{1+\sqrt[3]{2}}$ as a number $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ with fractions $a,b,c \in ...
9
votes
2answers
274 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
9
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2answers
661 views

Every ideal of an algebraic number field can be principal in a suitable finite extension field

Let $K$ be an algebraic number field. Let $I$ be a non-zero ideal of the ring of integers $\mathcal{O}_K$ in $K$. By class field theory, there exists a finite extension(the Hilbert class field) $L$ of ...
9
votes
2answers
129 views

What is the minimum polynomial of $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$?

Inspired by a previous question what let $x = \sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6} = \cot (7.5^\circ)$. What is the minimal polynomial of $x$ ? The theory of algebraic extensions says the degree is ...
9
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2answers
351 views

Are rational numbers + numbers constructed with a root = algebraic numbers?

I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it. We can construct the rational numbers using the division ...
9
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2answers
517 views

Primes dividing the values of integer polynomials

Problem: Let $n$ be an integer and $p$ a prime dividing $5(n^2-n+\frac{3}{2})^2-\frac{1}{4}$. Prove that $p \equiv 1 \pmod{10}$. The polynomial can be re-written as ...
9
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2answers
227 views

$\arctan$ of a square root as a rational multiple of $\pi$

I know that if $x$ is a rational multiple of $\pi$, then $\tan(x)$ is algebraic. Is there a fairly simple way to express $x$ as $\pi\frac{m}{n}$, if $\tan(x)$ is given as a square root of a rational? ...
9
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2answers
92 views

Prove there are infinitely many primes in $\mathbb{Z}[i]$

I saw a proof online there are infinitely many primes in $\mathbb{Z}$. The Euler product let's us factor the harmonic series: $$ \prod \left( 1 - \frac{1}{p} \right) = \sum \frac{1}{n}$$ I wonder ...
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1answer
1k views

Reference book for Artin-Schreier Theory

The aim of the question is very simple, I would like to study Artin-Schreier Theory, but I have had embarassing difficulties in finding a book which could help me in doing that. In specific I'm ...
9
votes
1answer
593 views

Computing the ring of integers of a number field $K/\mathbb{Q}$: Is $\mathcal{O}_K$ always equal to $\mathbb{Z}[\alpha]$ for some $\alpha$?

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
9
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2answers
279 views

Relationship between different L-functions

What's the relationship between between Artin $L$-functions and Dirichlet or Hecke $L$-functions if $L/K$ is an abelian extension? I've been told that one can interpret the Artin $L$-functions as ...
9
votes
4answers
147 views

Prove the series has positive integer coefficients

How can I show that the Maclaurin series for $$ \mu(x) = (x^4+12x^3+14x^2-12x+1)^{-1/4} \\ = 1+3\,x+19\,{x}^{2}+147\,{x}^{3}+1251\,{x}^{4}+11193\,{x}^{5}+103279\, ...
9
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1answer
165 views

Given a group $G$, does there exist a domain $D$ with $G$ as its ideal class group?

I have only recently encountered algebraic number theory and was wondering if this is the case. If the answer to the question is yes, then can we explicitly construct the domain $D$ ? Since the ...
9
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1answer
742 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 ...
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2answers
90 views

On discriminants and nature of an equation's roots?

Edited: All equations in the post are assumed to have all real coefficients and are minimal polynomials. While trying to ascertain if the Brioschi quintic $B(x)=x^5-10cx^3+45c^2x-c^2=0$ could ever ...
9
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1answer
168 views

Intuitive motivation to try to factor an ideal

In $\mathbb{Z} \sqrt{- 5}$, $2$ is irreducible, but the ideal $(2)$ factors into non-units: $(2)$ = $(2, 1 + \sqrt{- 5})(2, 1 - \sqrt{- 5})$. In general, what gives one the intuitive motivation (or ...
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2answers
901 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 ...
9
votes
1answer
547 views

$f'/f\in\mathbb{Z}[[x]]$ for polynomials vs. formal power series $f$

I am curious about the following problem from MIT's Problem Solving Seminar (#26 here, though the link may stop working after a few weeks): Let $f(x) = a_0+a_1x+\cdots\in\mathbb{Z}[[x]]$ be a ...
9
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1answer
225 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 ...
9
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1answer
279 views

$\mathbb{Q}(i)$ has no unramified extensions

It is a classical result that every extension of $\mathbb{Q}$ is ramified. Put differently: there are no unramified extensions of $\mathbb{Q}$. The classical proof follows from the following two ...
9
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1answer
799 views

Special cases of the Stark-Heegner theorem with simple proofs

The Stark-Heegner theorem states that the ring of integers of the quadratic number field $\mathbb Q(\sqrt{m})$, where $m$ is a squarefree negative integer, is a principal ideal domain, iff ...
9
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1answer
455 views

Intuition for Krasner's Lemma

From Milne's Algebraic Number Theory, we have (he assumes that $K$ is complete with respect to a discrete nonarchimedian absolute value, but I don't know where the discrete part is being used) Let ...
9
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2answers
357 views

What are some equivalent statements of (strong) Goldbach Conjecture?

What are some equivalent statements of (strong) Goldbach Conjecture ? We all know that Riemann Hypothesis has some interesting equivalent statements. My favorites are involved with Mertens ...
9
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2answers
131 views

Showing that a real number is an algebraic integer

For what values of $x,y,z\in\mathbb{Z}$, such that $0\leq x,y,z\leq 2, $ the real number $$\alpha:=\frac{1}{3}\left(x+\sqrt[3]{175} \cdot y+\sqrt[3]{245}\cdot z\right)$$ is an algebraic integer i.e. ...
9
votes
2answers
474 views

Exact power of $p$ that divides the discriminant of an algebraic number field

I am doing Marcus problem 21 (b) of chapter 3. The setup for this problem is given in problem 20: Setup: Let $L/K$ be a finite extension of algebraic number fields. Write $R = \mathcal{O}_K$ ...
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0answers
105 views

Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
9
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0answers
672 views

A proof of a theorem on the different in algebraic number fields

Theorem Let $K ⊂ L ⊂ E$ be a tower of algebraic number fields. Suppose that $E/K$ is a Galois extension. Let $B$ and $C$ be the rings of algebraic integers in $L$ and $E$ respectively. Let $G$ be the ...
8
votes
5answers
495 views

Applications of class number

There is the notion of class number from algebraic number theory. Why is such a notion defined and what good comes out of it? It is nice if it is $1$; we have unique factorization of all ideals; but ...
8
votes
5answers
235 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 $$5 = \left( \frac{7 - \sqrt{29}}{2} \right) \left( \frac{7 + \sqrt{29}}{2} \right)?$$ ...
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 ...
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2answers
484 views

Computing the Hilbert class field

Does anyone know any good source with nice examples of Hilbert class field computations? I'm trying to piece together the theory with some canonical examples.
8
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1answer
389 views

Any resource of the applications of the theory of class fields

We all agree that the theory of class fields plays an eminent role in modern number theory. Nevertheless, what was our main concern is that how to solve various Diophantine equations to which the ...
8
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3answers
300 views

Solve $x^3+x \equiv 1 \pmod p$

Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions. We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$ We can prove that $(1)$ cannot ...
8
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2answers
1k views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
8
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2answers
446 views

Classgroup of $\mathbb{Q}(\sqrt{2},\sqrt{-13})$

How would you compute the classgroup of the biquadratic number field $\mathbb{Q}(\sqrt{2},\sqrt{-13})$? I would prefer a method as "from scratch" as possible. Please avoid, if possible, quoting ...
8
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2answers
947 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
8
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2answers
2k views

what does it mean for a prime at infinity to ramify?

I understand what it means for a prime number to ramify in a ring of integers of a number field. However, an infinite prime is an archimedean valuation, what does it mean for an archimedean valuation ...
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3answers
525 views

Attaining the norm of an ideal in a number field by the norm of an element

Let $K$ be a number field of degree $n$ and $\mathfrak{a}$ be an ideal in its ring of integers $\mathcal{O}_K$. We can consider: The norm $N(\mathfrak{a})$ of $\mathfrak{a}$. The norms $N(x)$ of the ...
8
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2answers
414 views

A simple question about Iwasawa Theory

There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I ...
8
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2answers
209 views

Elements of cyclotomic fields whose powers are rational

Suppose the polynomial $t^k - a$ has a root (hence splits) in $\mathbb{Q}(\zeta_k)$. For which $k$ does it follow that one of the roots of $t^k - a$ is rational? In particular, are there infinitely ...
8
votes
3answers
259 views

Can the argument of an algebraic number be an irrational number times pi?

This is mainly out of curiosity. Let $\nu$ be an algebraic number. Can Arg($\nu$) be of the form $\pi \times \mu$ for an irrational number $\mu$?
8
votes
2answers
314 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 ...
8
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2answers
724 views

Class field theory for function fields and a curious statement

Let $X_0$ be a smooth curve over a finite field $\mathbb{F}_q$, and let $X$ be the base-change to the algebraic closure. I read that, according to class field theory in function fields, "the image ...
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2answers
838 views

How to find the integral closure of $\mathbb{Z}_{(3)}$ in the field $\mathbb{Q}(\sqrt{-5})$?

Let $v$ be the 3-adic valuation on $\mathbb{Q}$ and consider the subring $\mathbb{Z}_{(3)}$ of $\mathbb{Q}$ defined by $$ \mathbb{Z}_{(3)} = \{ x \in \mathbb{Q} : v(x) \geq 0 \}. $$ That is, ...
8
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2answers
417 views

What information do we gain from PIDs

I am self-learning some algebraic number theory and my question is regarding the advantages to studying PIDs. I have seen that Euclidean Domains $\subseteq$ Principal Idea Domains $\subseteq$ Unique ...
8
votes
3answers
1k views

Preparations for reading Algebraic Number Theory by Serge Lang

I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
8
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4answers
583 views

Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$

I want to prove the assertion: The unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ when $p \equiv 1 \pmod{4}$, respectively $\mathbb{Q}(\sqrt{-p})$ when $p \equiv 3 ...