Questions related to the algebraic structure of algebraic integers

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6
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1answer
118 views

$\mathbb{Q}(\sqrt[3]{17})$ has class number $1$

Let $\alpha:=\mathbb{Q}(\sqrt[3]{17})$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$ I have to show that $K$ has ...
7
votes
1answer
45 views

Product of “Fake”-Galois Conjugates

My apologies if this question ends up being a duplicate; I did my best to search for an answer, but I have no idea what to call this stuff I'm working with, so I couldn't really find much. There is a ...
0
votes
1answer
29 views

What is the intersection of $\mathbb Z$ with the ideal generated by $1-\zeta_n$?

For example, 1-(-1) is in the ideal <2>, whenever $n$ is even. Suppose $R=\Bbb Z[\zeta_n]$, and (by the above), that $n$ is odd. We know 1+$\zeta_n$ can be multiplied by ...
0
votes
1answer
47 views

Rational points on $4x^5 + y^2 = z^2$

Does the title curve have any nonzero rational points ? I have to admit that i didn't find any significant insight to this problem.
0
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0answers
30 views

What are the pre-requisites required to understand Milnor's book on algebraic K- theory?

I want to understand Steinitz’ theorem on the structure of finitely generated modules over Dedekind domains. I also want to have some general awareness regarding what Algebraic K-theory is about. ...
4
votes
1answer
87 views

Square and cubic roots in $\mathbb Q(\sqrt n)$

Here is my question : Let $n$ a squarefree positive integer, $m \ge 2$ an integer and $a+b \sqrt n \in\mathbb Q (\sqrt n).$ What (sufficient or necessary) conditions should $a$ and $b$ satisfy so ...
2
votes
1answer
47 views

P-adic expansion of rational number

Maybe this is a silly question but I really can not see how to get a p-adic expansion of a rational number. I do know the case of for an integer but how can I extend to the rational number case. If we ...
2
votes
1answer
40 views

How are unramified extensions of number fields formed?

Every extension is formed by adjoining a root of a polynomial. E.g.: Totally ramified = root of Eisenstein polynomial. Unramified over a local field = root of cyclotomic polynomial. What about ...
2
votes
2answers
27 views

Units become powers when lifted to unramified extensions?

Suppose $k$ is an algebraic number field, and $K$ is an unramified extension. I know: non-units $p\in k$ cannot become a power in $K$, or else the ideal they generate would become ramified in ...
1
vote
0answers
17 views

Volume of a convex body

Let $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$ and define the linear forms $L_i(x) = \sum_{j=1}^{n}a_{ij}x_j$ where $a_{ij} \in \mathbb{R}$. Define the domain $C$ by $$C : \{x \in \mathbb{R}^n : ...
2
votes
0answers
41 views

When is $\mathbb{Z}[\theta]$ a Dedekind domain for an algebraic number $\theta$?

The title says all. Suppose that $f\in \mathbb{Z}[t]$ is an irreducible polynomial (over $\mathbb{Q}$) and $\theta$ is a root of $f$. Can we determine when is $\mathbb{Z}[\theta]$ is a Dedekind ...
1
vote
1answer
19 views

Factorization of extension is injective

Let A be a Dedekind domain with field of fractions K, and let B be the integral closure of A in a finite separable extension L of K. Now I want to show the map from Id(A) to Id(B) is injective. I know ...
3
votes
2answers
50 views

Localising Dedekind domains

I'm wondering if the following is true: Let $A\subset B$ be two Dedekind domains with $B$ integral over $A$. Let $Q$ be a non-zero prime ideal in $B$ and $P=Q\cap A$. Then the localisation of $B$ ...
3
votes
3answers
63 views

Which ideal contains which? Or are they the same?

If I'm understanding this correctly, $13$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{-23})}$ is irreducible but not prime. From A & W we see that $x^2 \equiv -23 \bmod 13$ has solutions and therefore ...
4
votes
3answers
88 views

How can I integrate this? (for calculate value of L-function )

I want to calculate the definite integral: $$ \int_{0}^{1} \frac{x+x^{3}+x^{7}+x^{9}-x^{11}-x^{13}-x^{17}-x^{19}}{x(1-x^{20})}dx. $$ Indeed, I already know that $\int_{0}^{1} ...
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 ...
0
votes
1answer
23 views

Help to check a proof about local prime ideal being principal?

Let $K$ be a number field, call it's ring of integers $\mathcal O_K$ and take a - possibly nonprincipal - prime ideal $\mathfrak q$. I have shown that $\mathcal O_K$ is Noetherian integral domain and ...
1
vote
1answer
45 views

Matrix Algebra over Algebraically Closed Field

In Maclachlan and Reid's The Arithmetic of Hyperbolic 3-Manifolds, when proving that quaternion algebras are simple, they make use of the fact that $M_2(K)$, where $K$ is an algebraically closed ...
3
votes
1answer
56 views

algebraic conjugate

Let $\alpha, \beta$ be real roots of an irreducible polynomial over the field of rational numbers (i.e., $\alpha, \beta$ are algebraic conjugates). Is it possible that $\beta=\alpha^2$?
0
votes
1answer
28 views

$K|\mathbb{Q}_p$ un-ramified if and only if $(d_K)=(1)$: help with a passage

I need some help in th last passage of this proof: Suppose $K|\mathbb{Q}_p$ is un-ramified and of degree $n$. then $K=\mathbb{Q}_p(\alpha)$, where $\alpha$ can be taken to be an integral unit in ...
2
votes
0answers
19 views

Un-ramified extension of $\mathbb{Q}_p$. A clarification on the construction

I'm following the proof given in Koblitz's book which roughly speaking builds the un-ramified extension of degree $f$ of $\mathbb{Q}_p$ as $\mathbb{Q}_p(\alpha)$, where $\alpha$ is a root of the lift ...
3
votes
0answers
30 views

For what integer values of n is $\tan (\pi /n)$ an algebraic integer?

In http://oberlin.edu/faculty/jcalcut/arctan.pdf Calcut implies that this is true except when n is of the form $2{{p}^{k}}$for p an odd prime and k a natural number. He shows earlier that $\tan (\pi ...
2
votes
1answer
36 views

Compute Takagi group of the extension $\mathbb Q(i,\sqrt{-5})/\mathbb Q(\sqrt{-5})$

Given an extension $L/K$ of number fields we define the Takagi group as the subgroup $$T_{L/K} = N_{L/K} (D_L) \cdot H_K \subseteq D_K$$ where $N_{L/K}$ is the relative norm, $D_\bullet$ is the ...
3
votes
2answers
52 views

Dirichlet density for number fields $K$?

Let $K$ be a number field. Let $P$ be a subset of the set of nonzero prime ideals in $K$. For $\mathfrak{p} \in P$, let $N(\mathfrak{p})$ be its absolute norm, so $N(\mathfrak{p}) = ...
3
votes
1answer
40 views

Higher ramification groups of Galois extension of order $p^2$

Let $p\in \mathbb{Z}$ be a prime number and $K/\mathbb{Q}$ be a Galois extension of degree $p^2$ over $\mathbb{Q}$. Suppose that $P\subset \mathcal{O}_K$ is the only prime ramified over $p$. Let ...
4
votes
0answers
44 views

The group defined by Gauss's definition of composition of forms

In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. ...
2
votes
0answers
87 views

Ideal theoretic proof of the first inequality of global class field theory

In the old days(namely in the 1920s), global class field theory was stated and proved without using $\mathfrak p$-adic fields. I am interested in their methods, but unfortunately they were written in ...
2
votes
2answers
115 views

Is $\mathbb{Z}\left[\frac{1+\sqrt{-5}}{2}\right]$ a Euclidean domain?

I was reading some stuff on quadratic integers rings, and it seems like $R =\mathbb{Z}\left[\frac{1+\sqrt{-5}}{2}\right]$ is not supposed to be a Euclidean domain. But I got this: If $a,b \in R$, ...
2
votes
1answer
61 views

A subfield of $\mathbb{Q}(\zeta_{pq})$ with some ramification conditions

Let $p, q\in \mathbb{Z}$ be distinct primes such that $p$ splits into $r$ distinct primes in $\mathbb{Q}(\zeta_q)$ $\mathbb{Q}(\zeta_q)$ contains a subfield $F$ of degree $rf$ over $\mathbb{Q}$ ...
1
vote
0answers
25 views

How can I choose $\frak p\unlhd\cal O$ prime so $u\in\cal O^\times$ becomes a $n$-th power (mod $\frak p$)?

$k$ is an algebraic number field, and $\cal O$ is the ring of integers, $\cal O^\times$ is the set of invertible elements of $\cal O$. Suppose $u\in\cal O^\times$ is not a $n$-th power. How can I ...
3
votes
1answer
71 views

$3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$

I came across this problem: Prove there arent't any $a$, $b$ integers that satisfy equation $3ab + a^3 - 2b^3 - 4a + 5b - 7 = 0$ Firstly, I've thought something like this: $$(a^3 + b^3)-3b^3 ...
3
votes
1answer
27 views

The size of a particular subset of $\mathcal{O}_K$

Let $\mathcal{O}_K$ be the ringer of integers of a number filed $K$ of degree $n$ over $\mathbb{Q}$, with integral basis $\{\omega_1, \ldots, \omega_n\}$. Let $\mathfrak{a}$ be an integral ideal of ...
3
votes
0answers
35 views

How to write the Adeles over $\mathbb{Q}(i)$?

What do these adeles look like over $\mathbb{Q}(i)$? These are simply fractions where the numerator and denominator are allowed to have the number $i = \sqrt{-1}$. The elements look like: $$ ...
4
votes
0answers
39 views

Prove that in the ring of integers of a number field: a non-zero ideal coprime to $c\in\mathbb{Z_{\geq1}}$ has norm coprime to $c$

Let $\mathfrak{a}$ be a non-zero ideal of the RoI of a number field $A$, to say it is coprime to $c\in\mathbb{Z_{\geq1}}$ means: $\mathfrak{a}+cA=A$. The norm of $\mathfrak{a}$, or $N\mathfrak{a}$, ...
1
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0answers
33 views

Continuous maps from an absolute Galois group

Let $\xi$ be a continuous homomorphism from an absolute Galois group $G_{\bar{K}/K}$ (Krull topology) to a finite abelian group $M$(discrete topology), where $K$ is a number field and $\bar{K}$ is its ...
3
votes
2answers
64 views

Proving that the group of all roots of unity in a number field is finite cyclic

I would be very grateful if someone would check my solution to the following problem. Let $K$ be a number field (i.e. a finite field extension of $\mathbb{Q}$). Let $G$ be the group of all roots of ...
1
vote
0answers
56 views

Is solution to $\Gamma(x+1)=121$ algebraic?

If I had the following: $$x!=\Gamma(x+1)=121$$We see that $x\approx5$. But is the exact value of $x$ algebraic? For some non-whole number $x$ input that is algebraic, I think the output of the Gamma ...
3
votes
1answer
92 views

Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...
1
vote
1answer
40 views

$\mathcal{O}_L$ free over $\mathcal{O}_K[G]$

Let $L/K$ be a galois extension of number fields. Suppose $G:=\text{Gal}(L/K)$ is abelian. If $\mathcal{O}_L$ is free as $\mathcal{O}_K[G]$-module, is it true that it has rank 1?
3
votes
1answer
44 views

Which primes ramify when adjoining roots of a unit?

1) For ANF $K$, if $\zeta_n, u\in\cal O_k^\times$, are there any primes that ramify in $k(\sqrt[n]{u})/k$? 2) is the HCF composed solely of all such extensions, or are there others? Long story: Let ...
2
votes
1answer
33 views

What does $H(\mathcal O_K)[2]$ mean?

I'm currently studying some theory related to an imaginary quadratic field $K$. The definition of $H(\mathcal O_K)$ is the ideal class group of $K$. The notation in the title came up in the following ...
3
votes
0answers
52 views

About prime ramification in cyclotomic fields.

I got a question about lemma 2.2, in the appendix of Cyclotomic Fields I+II written by Karl Rubin. Let me explain its situation: a. $m$ and $M$ are fixed positive integers(especially, $M$ is odd). ...
3
votes
1answer
61 views

Is $\mathbb{Q}(\zeta_5)/\mathbb{Q}(\sqrt5)$ the maximal finite abelian extension of $\mathbb{Q}(\sqrt5)$ unramified away from $5\infty$?

The following problem appears in a homework question posed by B. Conrad (2(i) here: http://math.stanford.edu/~conrad/249BPage/homework/hmwk9.pdf): Using class field theory, prove that ...
3
votes
0answers
31 views

Is there something wrong in this proof of a factoring theorem?

In Murty and Esmonde's Problems in Algebraic Number Theory, a proof of the following theorem is given. Let $K$ be a finite degree extension of $\mathbb{Q}$, and $\mathcal{O}_K = ...
6
votes
1answer
163 views

Number of solutions to exceedingly contrived congruence.

Let $a$ be the number of solutions to$$x^{2011}-96x^{728}-x^{24}+67 \equiv y^{2011}+12718253987182795172957215781251235234235y \pmod{2^{57885161}-1}$$where $x$ and $y$ are integers in-between $0$ and ...
2
votes
3answers
58 views

Comparing two quadratic number fields

Let $\;$$K = \Bbb Q$[$\sqrt{-5}$] $\;$and let$\;$ $L = \Bbb Q$[$\sqrt{-6}$]$\,$. While it is clear that, as fields, K and L are distinct, if each arithmetic operation is considered separately, they ...
9
votes
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 ...
3
votes
0answers
27 views

Definition for Shimura datum

The following definition for $\textbf{shimura datum}$ is due to wikipedia. Let $S=\mathrm{Res}_\mathbb{R}^\mathbb{C}G_m$ be the Weil restriction of the multiplicative group from complex field ...
1
vote
0answers
23 views

Integral elements form a ring. What can we say about polynomials of sum and product? [duplicate]

Let $B$ be a ring (commutative and with identity). It is a standard fact in Algebraic Number Theory that the sum $b_{1}+b_{2}$ and the product $b_{1}b_{2}$ of integral elements $b_{1},b_{2}\in B$ over ...
3
votes
1answer
58 views

Group cohomology or classical approach for class field theory?

First of all, I don't think this is a duplicate, because the related questions I found were mainly about history of group cohomology in number theory and there was no one asking about the classical ...