Questions related to the algebraic structure of algebraic integers

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2
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1answer
53 views

Average of sum of unit roots is an algebraic integer

Let $\alpha_1,\ldots, \alpha_n$ be roots of unity, and let $a=\frac{1}{n}\sum\alpha_i$. Then if $a$ is an algebraic integer, we have either $a=0$ or $a=\alpha_1=\dots=\alpha_n$. Why?
2
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2answers
45 views

Why is the group of principal units of a local field uniquely divisible by $n$?

I am reading a proof with the followings setup and claim. $K/F$ is a Galois extension of local fields with group $G$ of order $n = q^s$, where $q$ is prime and $s \geq 1$. Assume the maximal ideal ...
0
votes
1answer
35 views

How do I prove this Diophantine equation has no solutions?

Let $p$ be an odd prime number. Prove that $x^2+2y^2=pz^2$ has no solutions in natural numbers with $x, y, z$ pairwise coprime and $y$ even unless $p\equiv 1$ (mod 8). I don't understand how to ...
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0answers
13 views

Why $d^{p} =0$,and $bdd^{p-1}$=0 and $^{p-1}$ is a derivation?

I am looking example in the article:$$$$ Example:Supoose that $K$is a field of char$K=p>2$ and $L$ is its field extension $L=K(b),b\not\in K,b^{p} \in K$.Let $d$ be a $K$-linear derivation of L ...
0
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0answers
27 views

Constants in Siegel's Lemma

I've got a hopefully straightforward question to ask concerning the following lovely version of Siegel's Lemma: Let $K$ be some algebraic number field, and let $O_K$ be it's ring of integers. Define ...
1
vote
1answer
32 views

Localisation and fractional ideal

I read this in Algebraic Number Theory by A. Fröhlich & M. J. Taylor on p94: $\mathfrak o$ is a Dedekind domain with field of fraction $K$. Let $L, M$ be finitely generated torsion free ...
1
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1answer
39 views

$B_{\mathfrak p}$ not always a simple extension of $A_{\mathfrak p}$?

Let $B$ be the integral closure of some ring of integers $A$ in an extension of number fields, and let $\mathfrak p$ be a prime of $A$. I've seen an example where $B$ is not a simple extension of ...
2
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0answers
40 views

Polynomials over the rationals with alternating group as Galois group.

Suppose $f(x) \in \mathbb{Q} [x]$ is irreducible, and of degree $n$. Let $K$ denote its splitting field over $\mathbb{Q}$, and let $G = \text{gal} (K : \mathbb{Q})$ be its Galois group. Now, if $G$ ...
0
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0answers
10 views

Inequalities on the field trace

Is there a list or a survey paper on inequalities on the field trace? More concretely, I'm interested in some problems of the following form: Are there a coprime pair of nonzero integer polynomials ...
3
votes
1answer
41 views

Let $p> 7$, prove that $\left(\frac{2}{q}\right) = (-1)^{\frac{q^2-1}{8}}$ with $q$ an odd prime

Let $p> 7$, prove that $\left(\frac{2}{q}\right) = (-1)^{\frac{q^2-1}{8}}$. with $q$ an odd prime. We can by using the following verifications: $$\left(\frac{2}{p}\right) = ...
2
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0answers
28 views

Hecke characters of imaginary quadratic fields

Let $K$ be a number field and let $\mathfrak m$ be an integral ideal. Let $I(\mathfrak m)$ be the group of fractional ideals of $K$ coprime with $\mathfrak m$. Let $P(\mathfrak m)$ be the group of ...
5
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0answers
86 views

For which norm-Euclidean domains is the proof interval less than $1$?

I saw a proof somewhere (can't find it at the moment) that shows $\textbf{Z}[\sqrt{-2}]$ is norm-Euclidean because for any pair of nonzero numbers $a, b$ it's possible to find a remainder $r$ such ...
2
votes
1answer
38 views

Why does $\mathfrak{p} \supseteq \mathfrak{a}$ imply $\mathfrak{p}$ is the only prime containing $\mathfrak{a}^{ec}$?

Let $\mathscr{o}$ be a one-dimensional noetherian integral domain (or more specifically, an order in an algebraic number field). Let $\mathfrak{a} \not= 0$ be an ideal of $\mathscr{o}$. Why does ...
3
votes
1answer
72 views

The ring of multipliers is an order

$\newcommand{\O}{\mathcal{O}}$ I am trying to solve Exercise 12.3 at page 84 in Neukirch, "Algebraic Number Theory". The exercise is following: Let $K$ be a number field of degree $n$. Let ...
2
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0answers
24 views

Unramified galois cohomology of roots of unity

Let $K$ be a local field. Let $n$ be a natural number which is not divisible by the residue characteristic and consider the maximal unramified extension $K_{nr}.$ I would like to show that ...
2
votes
1answer
54 views

Failure of flatness in an integer ring

In Lang's "Algebra" (chap 16, p 614) he states the following without proof: "Let $R$ be some ring in a finite extension $K$ of $\mathbb{Q}$, and such that $R$ is a finite module over $\mathbb{Z}$ ...
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0answers
40 views

What is the correct generalization of degree of a divisor to the number field case?

When describing smooth algebraic curves over a field $k$, there are (at least) two useful notions of "class group". The first generalizes easily to general schemes: the Picard group can be defined as ...
3
votes
1answer
55 views

Smallest positive integer in an ideal of a number field

Let $F$ be a number field and $I$ be a nontrivial ideal of the ring of integers. Show that the norm $N_{F/\mathbb Q}(I)$ has the same prime factors as the smallest positive integer in $I$. We ...
0
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1answer
30 views

Reconstructing formal groups from the p-map, realizing p-maps from formal group

Suppose $F$ is a formal group over $\mathbb{Z}_p$. There are few trivial condition that $f \in \mathbb{Z}_p[[t]]$, the power series representing the $p$-map should satisfy: 1)$f \equiv g(t^p) mod p$ ...
3
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1answer
38 views

How to compute gcd in real quadratic ring using the Euclidean algorithm

This may be correct or incorrect: $$\gcd(2, 1 + \sqrt{7}) = 3 + \sqrt{7}.$$ I got this by looking at prime factorizations: $$(3 - \sqrt{7})(3 + \sqrt{7}) = 2$$ $$(-3 - \sqrt{7})(2 - \sqrt{7}) = 1 ...
10
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2answers
258 views

Motivation on how does complex analysis come to play in number theory?

I am not sure if this is a appropriate question. If it isn't, let me know and I'll delete it. $\textbf{Background}$ I am an undergraduate student and I'm very interested in number theory. I've tried ...
1
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1answer
28 views

Relation between a maximal ideal and an invertible ideal

Let $D$ be a domain, $\mathfrak{a,b,p}\subsetneq D$ ideals with $\mathfrak{p}$ maximal and $\mathfrak{a}$ invertible (there is some $\mathfrak{c}$ ideal with $\mathfrak{ac}=\mu D$, with $\mu\in ...
1
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1answer
22 views

Quadratic extension. Decomposition of primes

I know the following fact from basic number theory. Let $K=\Bbb{Q}(\sqrt{d})$ be a quadratic number field. Let $p$ be a prime. Then the fact that there is only one prime $\mathfrak{P}$ above $p$ in ...
0
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1answer
17 views

An extension $w$ of a valuation $v$ induced from $\mathfrak{p}$ come from ideal $\mathfrak{q}$ above $\mathfrak{p}$?

Assume the standard context for extension of valuations. An extension $w$ of a valuation $v$ induced from a prime ideal $\mathfrak{p}$ comes from a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$? ...
4
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2answers
74 views

Integral closure of the p-adic integers in a finite extension of the p-adic numbers

In Cassels' article "Global Fields", he uses the term "ring of integers" and the notation $\mathcal{O}_K$, where $K$ is a field with a non-archimedean valuation, to denote the ring of elements $x \in ...
3
votes
1answer
100 views

What is the group of units of the localization of a number field?

Let $\mathcal{O}_K \subseteq K$ be the ring of integers of a number field. We have Dirichlet's Unit Theorem which says that the group of units of $\mathcal{O}_K$ is a finitely generated abelian group ...
2
votes
1answer
63 views

Is $\mathbb{Z}[\sqrt[3]{2}]$ a principal ideal domain? [closed]

Is $\mathbb{Z}[\sqrt[3]{2}]$ a principal ideal domain? That is, is every ideal of $\mathbb{Z}[\sqrt[3]{2}]$ generated by a single element?
1
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1answer
23 views

Two decomposition groups. Are they the same?

I am assuming the usual framework and notation of ramification theory. Let $G=\operatorname{Gal}(L/K)$. We define the decomposition group of a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$ as ...
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0answers
160 views

Class field theory for $p$-groups. (IV.6, exercise 3 from Neukirch's ANT.)

I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let $G$ be a profinite ...
4
votes
1answer
27 views

Infinitely many pairwise congruent solutions modulo $c$?

The following is from my number theory textbook.. In particular, there exists an integer $c$ such that there are infinitely many solutions to the equation $x^2 - dy^2 = c$. Since there are only a ...
2
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0answers
68 views

Second Course in Algebraic Number Theory - Lang versus Neukirch

So the title pretty much says it all. I have completed a first course in Algebraic Number Theory (number fields, ideal factorization in the ring of integers, finiteness of the ideal class group, ...
3
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0answers
16 views

$q_I$ primitive as a quadratic form? [closed]

Let $I \subset \mathcal{O}_k$ be an ideal, $N(I) = [\mathcal{O}_K : I] = |\mathcal{O}_K/I|$. Define $q_I$ be $q_I(x) = N_{K/\mathbb{Q}}(x)/N(I)$. Is $q_I$ primitive as a quadratic form?
3
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2answers
42 views

Ideals which contains an element

Let $\theta=\dfrac{1+\sqrt{-31}}{2}$, determine which ideals of $D=\mathbb{Z}[\theta]$ contains $1+\theta$. I know that if i.e $6\in\mathfrak{a}\Rightarrow \mathfrak{a}\mid 6D$ and then ...
2
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1answer
34 views

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}$, special cases.

What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}$$in the following two cases? When $ab \neq 0$ and $p = 2$. When $ab = 0$.
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0answers
108 views

Arithmetic Derivative

In Calculus, whenever we see a constant and want to take the derivative of it, it always is $0$. However in Number Theory, we have something called the arithmetic derivative in which we can ...
4
votes
1answer
49 views

Expressing an element in a number field as a ratio of two coprime integers

Let $K$ be a number field and $O_K$ be its ring of integers. Take $\alpha \in K$ and consider the principal fractional ideal $(\alpha)$. I am sure the following is true, but I couldn't quite prove ...
0
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0answers
44 views

Showing that a certain group of a number field is dense in $\mathbb{C}^*$

Let $K$ be an imaginary quadratic field, and $O_K$ to be its ring of integers. Let $M$ be a non-zero ideal of $O_K$. In an passage I am reading (related to Hecke characters), it states: the group ...
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2answers
54 views

What is the minimal polynomial of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$? [closed]

What is the minimal polynomial over $\mathbb{Q}$ of $\bigotimes_{j=0}^{\infty}\mathbb{Q}(\zeta_{j})$, where $\zeta_j$ is a $j$-th primitive root of unity for each $j$? I want to say it should be ...
0
votes
1answer
22 views

Prime that splits but not completely in $\mathbb{Q}(\zeta_{20})/\mathbb{Q}$

I need a prime that splits but not completely in $\mathbb{Q}/(\zeta_{20})\mathbb{Q}$. Is there any way to find one quickly?
1
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1answer
41 views

Question regarding the definition of different of a number field

Let $K$ be a number field. I was getting a bit confused, because different sources I looked into had different definitions. I was wondering if they were equivalent or not. And if they are how can I ...
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0answers
43 views

Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is ...
3
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0answers
37 views

locally algebraic representations

Let $K$ be a number field. Consider $$ \rho_{\ell}: \mathrm{Gal}(\bar K/K) \longrightarrow \mathrm{GL}(V)$$ an $\ell$-adic Galois representation. Assume it is semi-simple rational and abelian. Is ...
2
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0answers
24 views

Density of integers that are norms of ideals for $K \ne \mathbb{Q}$

I am interested in proving and understanding the following statement: If $K \ne \mathbb{Q}$, then the set of positive integers that are norms of ideals in $\mathcal{O}_K$ have density zero in ...
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0answers
13 views

rank of $\mathcal{U}(\Bbb{Z}[\zeta_5])$

I have this doubt while studying Dirichlet unit theorem. By dirichlet unit theorem we know that $\mathcal{U}(\Bbb{Z}[\zeta_5])=C\times F$ where $C$ is finite cyclic and $F$ is free abelian of rank ...
3
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1answer
56 views

Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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0answers
28 views

Power residue symbol

Let $\mathscr{P}\subset\mathbb{Z}[\zeta_m]$ be a prime ideal of norm $p$, and consider the power residue symbol $$\psi(x)= \left( \frac{x}{\mathscr{P}} \right)_m.$$ Let $g$ be a primitive root modulo ...
0
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1answer
31 views

Ramification of prime ideal in Kummer extension

Let $\mu \in \mathbb{Q}(\zeta_n)$ lie above the rational prime $p\equiv 1 \text{ mod }n$, and let the prime ideal $\mathscr{P}\subset \mathbb{Z}[\zeta_n]$ have exponent $a$ in $(\mu)$. Why is it ...
2
votes
2answers
61 views

How to construct an n-gon by ruler and compass?

Since $\cos[\frac{2\pi}{15}] $ is algebraic and equal to $\frac{1}{8}(1+\sqrt{5}+\sqrt{30-6\sqrt{5}})$ we know that the regular 15-gon is constructible by ruler and compass. Although I know how to ...
0
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1answer
30 views

Polynomial rings which are Dedekind domains. [closed]

Let $K$ be a field. Is $K[X,Y]$ a Dedekind domain?
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0answers
32 views

Are there non-abelian totally real extensions?

I'm aware that abelian extensions are either totally real or CM. Also, Galois extension are either totally real o totally imaginary. But I'm wondering about the converses to those statements. Are ...