Questions related to the algebraic structure of algebraic integers

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5
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1answer
43 views

$L$-function absolutely convergent for $\text{Re}(s) > 1$, condition for $L(s, \chi)$ converging for $\text{Re}(s) > 0$?

I have two questions related to here. Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ...
4
votes
1answer
37 views

$|\chi(\mathfrak{a})| = 1$ for any ideal $\mathfrak{a}$?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...
6
votes
1answer
58 views

Product of two absolutely convergent Dirichlet series

We have$$(f * g)(n) = \sum_{d \mid n} f(d)g(n/d).$$How do I see that if the two Dirichlet series$$F(s) = \sum_{n =1}^\infty f(n)n^{-s},\text{ }G(s) = \sum_{n=1}^\infty g(n)n^{-s}$$converge absolutely ...
3
votes
0answers
48 views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
1
vote
2answers
26 views

Neukirch Definition of Map into $K_{\mathbb{C}}$

I'm confused about how Neukirch defines this $j$ map at the beginning of his section on Minkowski theory. Here is his definition: I don't really understand what ...
3
votes
1answer
68 views

$\langle 7, 3 + \sqrt{-5} \rangle = \langle 7, 4 + \sqrt{-5} \rangle$, right?

I'm not in a math class (haven't been in years) but if this question about $\textbf{Z}[\sqrt{-5}]$ appears in some textbook, I wouldn't be surprised. What I have done: $$1 \times 7 + (3 - ...
2
votes
2answers
35 views

Minimal polynomial of integral elements [duplicate]

Let $R$ be an integrally closed domain and let $K$ be its fraction field. Let $L\supseteq K$ be a field. If $\alpha\in L$ is integral over $R$ (i.e. if it satisfies a monic polynomial in ...
2
votes
0answers
27 views

Inertia group - local vs global field

In the case of a totally ramified extension of a complete field, then we know that there exist a maximum tamely ramified subextension, which is of degree coprime to $p$, and from that onward there are ...
1
vote
0answers
24 views

Compare Valuations

Let $K$ be a field complete with respect to a discrete valuation $v$. Let $K_s$ its separable closure, $w$ the unique valuation extending $v$, $(\mathcal{O}_{K_s}$, $\mathfrak{M})$ respectively the ...
3
votes
0answers
99 views

How to compute the integral closure of $\Bbb{Z}$ in $\mathbb Q(\sqrt[n]{p})$?

We have the definition of integral closure that all the integral elements of A in B. Could we just compute the integral closure of certain A in B. I am considering such a problem that given a prime p, ...
1
vote
2answers
57 views

Is $K=\mathbb{Q}(\sqrt{-5+\sqrt{5}}) $a Galois extension?

I am wondering if the number field $K=\mathbb{Q}(\sqrt{-5+\sqrt{5}})=\mathbb{Q}(\alpha)$ is a Galois extension. I think it is and I have the following argument, but it feels like a circulair argument: ...
9
votes
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 ...
3
votes
1answer
31 views

All open subsets of the spectrum of a number field are principal

Let $K$ be a number field, $\mathcal{O}_k$ its ring of integers, $\operatorname{Cl}(K)$ its class group and $h_K = \lvert \operatorname{Cl}(K)\rvert$ its class number. Let $X = ...
1
vote
2answers
58 views

Proof of Proposition 2.12 in Neukirch ANT

I'd like a reference or a direct proof of the following statement: Let $K|\mathbb Q$ be a finite extension and consider the ring of algebraic integers $\mathcal O_K$. Let $\mathfrak ...
6
votes
1answer
85 views

if $p\mathcal{O}_K$ splits completely in Galois extension, are all primes lying over $p$ generated by one class in $CL(K)$?

L.S., Studying for my exam on algebraic number theory, I was thinking of writing down tricks for computing the class group of a number field fast. I thought of the following one, but I don't know if ...
6
votes
0answers
104 views

Showing that an ideal class group is cyclic

Let $K=\mathbb Q(i,\sqrt{14})$ and show that $\operatorname{Cl}(K)$ is cyclic. Note: This is the first part in a four part question. The second part is to compute the ideal class group, so presumably ...
6
votes
1answer
36 views

$p$-adic logarithm is injective if $p > 2$?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i = 1}^\infty (-1)^{i-1}x^i/i.$$I know that $\log_p$ is a homomorphism from $U_1$ to the additive group of $\mathbb{Q}_p$, where $U_1$ is the ...
2
votes
0answers
31 views

There are infinite number of degree $1$ principal prime ideal in a ring of algebraic integers

Let $K$ be a number field. I was wondering how we know that there are infinite number of degree $1$ principal prime ideal of $K$. Context: This is related to an example of polynomial representing ...
5
votes
1answer
136 views

On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$

The equation, $$ (ax_1^2+by_1^2)(ax_2^2+by_2^2) = ax_0^2+by_0^2\tag1$$ has the well-known solution when $a=b=1$, $$ (x_1^2+y_1^2)(x_2^2+y_2^2) = (x_1 y_2 + x_2 y_1)^2 + (x_1 x_2 - y_1 y_2)^2$$ ...
0
votes
1answer
18 views

Ramification group definition in Neukirch's Chap. 2 Section 9

Let $L/K$ be an algebraic Galois extension and $w/v$ a non-archimedean extension of valuations. Let $\mathcal{O}$ and $\mathfrak{P}$ denote the valuation ring and valuation ideal of $(L,w)$. After ...
4
votes
1answer
64 views

$p$-adic logarithm, $|\log_p(1 + x)|_p = |x|_p$?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i =1}^\infty (-1)^{i-1}x^i/i.$$How do I see that if $p > 2$ and $|x|_p < 1$, then $|\log_p(1 + x)|_p = |x|_p$?
3
votes
1answer
69 views

Lack of Ramification in Cyclic Extensions

Notation/Setup. Let $p$ be a prime greater than 2 and $d$ be a square free integer such that $\gcd(d,p) = 1$. Question. Suppose we pick $d\equiv 3\mod 4$, in which case $\mathbf{Q}(\sqrt{d})$ is ...
1
vote
1answer
66 views

Computing $\mathcal{O}_K*$ by looking at the subfields

I have a question regarding some notes that my lecturer made on Algebraic Number Theory. We want to compute the unit group $\mathcal{O}_{K}*$ for the field $K=\mathbb{Q}(\sqrt{-2},\sqrt{3})$. We have ...
1
vote
1answer
57 views

Prove a property of the order of conductor $f$ in the field $\mathbb{Q}(\sqrt{D})$

Let $D$ be a squarefree integer, and let $\mathcal{O}$ be the ring of integers in the quadratic field $\mathbb{Q}(\sqrt{D})$. For positive integer $f$ define the order of conductor $f$, ...
1
vote
0answers
25 views

fractional ideals of a number field - can they be multiplied by an integer to land in $\mathcal{O}_K$

Let $K$ be a number field (i.e a finite extension of $\mathbb{Q}$) and $\mathcal{O}_K$ its ring of integers, and $I \subseteq K$ a fractional ideal, by which I mean/define an finitely generated ...
1
vote
1answer
36 views

Neukirch ANT I.9.4: Surjective morphism $G_\mathfrak{P} \to G (\kappa(\mathfrak{P})|\kappa(\mathfrak{p}))$

I think there is a gap in the proof of this proposition and am wondering how to fix it. $L|K$ is a Galois field extension, $\mathcal{O}$ and $\mathcal{o}$ their rings of integers, $\mathfrak{P}$ is a ...
2
votes
0answers
44 views

How many squares in a row can be found in a quadratic progression?

If we have a polynomial of degree $2$ with rational coefficients, it may or may not be a square of a degree $1$ polynomial. Say we look at $f(0),f(1),f(2),\ldots$ and observe that they are all ...
15
votes
1answer
335 views

Understanding proof by infinite descent, Fermat's Last Theorem.

See here. The question is as follows. How do we see that there do not exist nonconstant, relatively prime, polynomials $a(t)$, $b(t)$, and $c(t) \in \mathbb{C}[t]$ such that$$a(t)^3 + b(t)^3 = ...
0
votes
1answer
41 views

Understanding the proof of this theorem leading up to Dedekind's theorem

I am reading Murty & Esmonde's Problems in Algebraic Number Theory and was wondering if anyone can offer some clarification on the proof of this theorem: Let $p \in \mathbb{Z}$ be prime and ...
1
vote
0answers
16 views

Motivation for dual bases

I am encountering dual bases for the first time in the context of algebraic number theory, mainly in proofs regarding the existence of an integral basis for $\mathcal{O}_K$ and its ideals. I am ...
1
vote
1answer
49 views

Find polynomial to use for prime ideal factorization

L.S., In an exercise for my algebraic number theory homework I came across the following problem: I would like to factor ideals $(2)$ and $(7)$ in $K = \mathbb{Q}(i, \sqrt{14})$. I managed to show ...
2
votes
1answer
59 views

Divisibility of Differents in Tower of Fields

In Lemmermeyer's class field theory notes, on page 114 there is the following claim. Let $F = \mathbb Q(\sqrt{-5})$, then let $K = F(\sqrt{-1})$. Let $F_1 = \mathbb Q(\sqrt{-1})$ and $F_2 = \mathbb ...
1
vote
1answer
38 views

Proof that the Galois Field of order 8 is a field.

We know that the Galois Field of order 8 is isomorphic with $$\left( Z_2[x]^{< 3}, +_{d(x)}, \times_{d(x)}\right) $$ (Field of polynomials with coefficients in $Z_2$ and of grade smaller than 3, ...
6
votes
1answer
60 views

Property of Dirichlet character

Let $m$ be an integer prime to $p$ such that $\chi^m = \chi_0$ on elements of $\mathbb{F}_p^\times$. We let $\zeta_m$ be a primitive $m$th root of unity. For $b$ any integer prime to $m$ define ...
6
votes
3answers
96 views

Showing that a cubic extension of an imaginary quadratic number field is unramified.

Let $\alpha^3-\alpha-1=0$, $K=\mathbb Q(\sqrt{-23})$, $K'=\mathbb Q(\alpha)$, and $L=\mathbb Q(\sqrt{-23},\alpha)$. Then I am asked to show that the field extension $L/K$ is unramified. I know that ...
1
vote
2answers
15 views

Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
5
votes
1answer
105 views

Idea behind the definition of different ideal

Let $L/K$ be an extension of number fields. Let $I$ be a fractional ideal in $L$ and $$I^*:=\{x\in L \mid \text{Tr}_{L/K}(xI)\subset \mathcal{O}_K\}.$$ The different of $I$ is the following fractional ...
9
votes
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 ...
10
votes
1answer
152 views

Integer solutions of $x^3+y^3=z^3$ using methods of Algebraic Number Theory

I'm asked to prove that the famous equation $$x^3+y^3=z^3$$ has no integer (non-trivial) solutions, i.e. FLT for $n=3$ I'm aware that on this website there are solutions using methods of Number ...
2
votes
2answers
91 views

Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.

Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic. attempt: Suppose $\alpha, \beta \in \mathbb{C}$ such ...
4
votes
1answer
35 views

What is the Strategy in Computing Ideal Class Number?

I found many examples on computing ideal class numbers, but none gave an explicit statement on what we are examining when we are running through a list of elements with their norms written out. The ...
4
votes
1answer
72 views

What is this notation? $V(\Bbb Z/p\Bbb Z)$

I'm trying to write a blog post, and I've run into a stumbling block with notation. Is $V(\Bbb Z/p\Bbb Z)$ a standard notation in algebraic number theory? Does it mean a variety restricted to the ...
1
vote
1answer
66 views

Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3? [duplicate]

We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of ...
0
votes
1answer
30 views

Confusion about proof of necessary + sufficient condition for $\theta \in \mathbb{C}$ to be an algebraic integer

I am confused about a step in a proof of the following statement in Stewart Tall's ANT book: $\theta \in \mathbb{C}$ is an algebraic integer if and only if the additive group $G$ generated by powers ...
4
votes
2answers
100 views

Given $p \equiv q \equiv 1 \pmod 4$, $\left(\frac{p}{q}\right) = 1$, is $N(\eta) = 1$ possible?

Given distinct primes $p$ and $q$, both congruent to $1 \pmod 4$, such that $$\left(\frac{p}{q}\right) = 1$$ and obviously also $$\left(\frac{q}{p}\right) = 1$$ is it possible for the fundamental unit ...
1
vote
1answer
38 views

No ideals coprime in DVR?

There's this exercise in Neukirch, chapter I, §3 (i've changed the statement to deal only with the case that bothers me): Let $\mathfrak o$ be a Dedekind domain and $\mathfrak m$ be a nonzero ...
5
votes
2answers
86 views

Problem with units in number field

Edit:There were several major mistakes by my side this post, most of which have been accounted for.Now, after editing these out, the post seems to have no purpose at all.Nevertheless, it feels wrong ...
2
votes
1answer
29 views

Relation between a quadratic residue and it's order

In the context of the multiplicative group $(\mathbb{Z}/m\mathbb{Z})^\times$ of congruence classes modulo $m$ coprime with $m$, is there a theorem that states something about the order of a given ...
7
votes
1answer
58 views

Variant of strong approximation.

Let $K$ be a global field. Let $w$ be a place of $K$. Let $\textbf{A}^w$ be the restricted direct product over all $v$ except $w$ of the $K_v$ with respect to the subgroups $\mathcal{O}_v$. How do I ...
1
vote
1answer
20 views

Is $\widehat{K}L$ complete?

Let $K$ be a field and $\widehat{K}$ be a completion with respect to some valuation on $K$. Let $L$ be a finite separable extension of $K$. When regarded as a subfield of $\widehat{L}$, is ...