Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

5
votes
1answer
76 views

Subgroups of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$

If I'm understanding the main theorem of (infinite) Galois theory correctly, applied to $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, it gives us: a) all its open subgroups are ...
1
vote
0answers
11 views

Regulator of number fields doesn't vanish

The regulator of a number field $K$ is usually presented at the beginning of books on algebraic number theory, alongside the class number group, Dirichlet unit theorem... But the only proof for the ...
2
votes
0answers
18 views

Upper Numbering of Ramification Groups of Absolute Galois Groups for Totally Ramified Extensions

Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois ...
2
votes
1answer
24 views

By evaluating $\sum_{t} (1+(t/p))\zeta^t$ in two ways, prove $g=\sum_{t} \zeta^{t^2}$

I would appreciate help, please, with Exercise 6.11 in "Ireland and Rosen" (self-study). By evaluating $(1)$ $\sum_{t} (1+(t/p))\zeta^t$ in two ways, prove $(2)$ $g=\sum_{t} \zeta^{t^2}$ ...
0
votes
1answer
33 views

Characters defined by cyclic extensions

Let $F$ be a finite cyclic extension of degree $p$ over ${\bf Q}$. As I understand it, there is a way to associate a cyclic character to this extension. How does one do this explicitly? And how far ...
1
vote
0answers
54 views

Value group of simple field extension: Are these value groups equal?

Suppose that a field extension $L/K$ is finite, K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ is ...
4
votes
1answer
50 views

Non-archimedean exponential valuation and integral closure

I am trying to solve the following problem from Neukirch's book on ANT: Let $L|K$ be a finite field extension, $v$ a nonarchimedean exponential valuation, and $w$ an extension to $L.$ If ...
2
votes
1answer
48 views

Questions that SAGE, MAGMA can answer?

I practice theoretical mathematics and I know (almost) nothing about SAGE, MAGMA. I would like to know (in general) what type of questions can I ask SAGE to do? For example, I know that given an ...
2
votes
1answer
45 views

Properties of the norm in a Euclidean Domain

I am aware of the fact that the Euclidean Norm does not need to be unique in a given domain, however my question is essentially: can we ensure that the properties of the norm remain the same? More ...
2
votes
1answer
519 views

On orders of a quadratic number field

Let $K$ be an algebraic number field of degree $n$. An order of $K$ is a subring $R$ of $K$ such that $R$ is a free $\mathbb{Z}$-module of rank $n$. Let $\alpha_1, \cdots, \alpha_n$ be a basis of $R$ ...
2
votes
0answers
45 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
0
votes
0answers
42 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
2
votes
1answer
45 views

zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : ...
5
votes
2answers
82 views

Can we describe nicely all the rational numbers of the form $x^{2}-xy+y^{2}$?

Let $\zeta$ be a primitive cubic root of unity, i.e., $\zeta$ is a complex number such that $\zeta^{3}=1$ and $\zeta\neq1$. Let us consider the Galois extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ (the ...
0
votes
0answers
50 views

A question about square roots of quadratic residues.

Suppose $\mathbb{Z}_p^*$ ($p$ is a prime) is a cyclic group with generator $g$. We consider a subgroup $\mathbb{G}$ of $\mathbb{Z}_p^*$ with generator $h$ and order $q$, where $h = g^4~mod~p$ and ...
0
votes
0answers
38 views

Non-existence of a particular type of tower of number fields

I have number fields $\mathbb{Q}\subset K\subset H$ where $K\subset H$ is Galois. I want to show that is is impossible for a rational prime $p\in\mathbb{Z}$ to remain first inert in $K$ but then for ...
3
votes
1answer
31 views

Determining when ring of integers is $\mathbb{Z}[\theta]$

Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
2
votes
0answers
74 views

What are some practical attempts to disprove Riemann Hypothesis?

Most people believe Riemann Hypothesis is true. Since RH has not been proved yet, so it is not completely insane to disprove RH. Among the ways to disprove RH, straightforward ways, such as: try to ...
2
votes
1answer
67 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 ...
1
vote
1answer
15 views

Confusion about definition of primitive polynomials

I am working through Neukirch's Algebraic Number Theory and am confused about his definition of primitive polynomials on page 129. He defines $f(x)=a_0+a_1x+\dots +a_nx^n$ on $\mathcal{O}$ with ...
1
vote
2answers
21 views

ramification of prime in Normal closure

Let $K$ be an algebraic number field and let $p$ be a prime in $\mathbb{Q}$ such that $p$ ramifies in $L$, the Galois closure of $K$. How can I show that $p$ ramifies in $K$ itself?
3
votes
1answer
116 views

Factors of the numbers of the form $a^2+nb^2$

Let $N=a^2+nb^2$ with $\gcd(a,b) =1$ and $n \in \mathbb{Z^+}$. If $N=xy$ where $x$ and $y$ are relatively prime numbers, in what condition can $x$ and $y$ be also written in the same form as $N$ ...
2
votes
0answers
86 views

Ray class field of $K=Q (\zeta_3)$ of conductor 6.

I am studying class fields generated as Kummer extensions and I have studied decomposition laws in Kummer extensions. My particular example is $L=K(\sqrt[3]2 )$, where $K$ contains primitive cube ...
3
votes
1answer
56 views

Unramified algebraic extensions of local fields

This is a basic question from Neukirch's Algebraic Number Theory, Prop. 7.2: Fix a non-Archimedean local field $K$. Let $L/K$ and $K'/K$ be two extensions inside an algebraic closure $\bar{K}/K$ and ...
0
votes
1answer
86 views

Constructing idele from a rational number.

I am a novice to concept of idele, despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance, suppose I take $q=396000$. Does it mean that the idele ...
0
votes
0answers
35 views

what is the volume of $cos\pi \theta$? [duplicate]

I wana prove that if a=cos$\pi\theta$ is rational number(and also assume $\theta$ is rational,too) it can be just {-1/2,1/2,1,-1,0}.Before this, I proved that a is an algebraic number and I know that ...
9
votes
1answer
98 views

$\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field.

The problem I'm facing is that of the tittle: Problem. Prove that $\mathbb{Q}(\sqrt{23})$ is not a Euclidean number field. Since $23\not\equiv 1\pmod{4}$, it must be shown that ...
2
votes
1answer
66 views

$X^n + X + 1$ reducible in $\mathbb{F}_2$

I was told that sometimes in characteristic 2 that $X^n + X + 1$ is reducible mod 2. What is the smallest $n$ where that is true?
2
votes
1answer
32 views

What are the units in $\mathbb{Z}[\root 3 \of 2]$?

I asked Wolfram Alpha to tell me the fundamental unit of $\mathbb{Z}[\root 3 \of 2]$, it replied $1 - \root 3 \of 2$. Then I tried asking it for $(1 - \root 3 \of 2)^n$ for $-5 \leq n \leq 5$. If I ...
2
votes
1answer
79 views

Transitivity-like Results in Group, Ring, Module, Field and Galois Theory [closed]

I am reading Michael Atiyah and Ian Macdonald's Introduction to Commutative Algebra. On page 28, Proposition 2.16 says: Suppose $A,B$ are rings, $N$ is a finitely generated $B$-module, $B$ is ...
3
votes
1answer
35 views

Show that $\xi^3\equiv \pm 1 \pmod{\lambda^4}$ in $\Bbb Z [\omega]$

We have $\lambda=1-\omega$ where $\omega=e^{i 2\pi/3}$ and $\xi$ an Eisenstein integer. Given that $\xi \equiv \pm 1 \pmod{\lambda}$, how can I prove that $$\xi^3\equiv \pm 1 \pmod{\lambda^4}$$ I ...
1
vote
2answers
34 views

When is norm surjective mod an ideal for global fields?

Given $K/k$ Abelian, for which ideals of $\frak p\unlhd\cal O_k$ will we have $N^K_k:\cal O_K\rightarrow\cal O_k/\frak p$ surjective? $k$ is an algebraic number field. In his article "On the norm ...
1
vote
1answer
25 views

Index of norm group of a global field

For a global field $K$ (characteristic $p$ or $0$), is there anything meaningful which could be said about the value $[K^\times:N_{L/K}L^\times]$ where $L$ is a finite extension (possibly of prime ...
6
votes
3answers
168 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 ...
1
vote
1answer
52 views

Field extension equality using Kronecker theorem

Kronecker's theorem says that a field extension can be shown as say, F(a) represented as F[x]/minimalpoly(a). Say, Q[$\sqrt{2}$]=Q[x]/$(x^{2}-2$) And a well known example is ...
1
vote
0answers
51 views

An problem of ideal splitting in number field extension

If $L/K$ is Galois extension of number field, $\mathfrak{p}$ is an prime integral ideal of $K$. One would asserts that: $\mathfrak{p}\mathcal{O}_L=\mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}$. ...
2
votes
0answers
40 views

Monotonic roots

Consider we have a stricktly increasing positive sequence $\lambda_n$ and the following sixth order algebraic equation for every $n\in \mathbb{N}$, $$\zeta s^6-s^4+\lambda_n^2=0,$$ where $\zeta$ is a ...
3
votes
3answers
416 views

Field containing all square roots of rational numbers

What is the smallest field which contains all square roots of positive rational numbers? I guess I mean “smallest” in terms of set inclusion, i.e. the minimal one with regard to the “$\subseteq$” ...
4
votes
1answer
98 views

Is $3$ prime in the ring of integers of the field $\mathbb{Q}(\sqrt{2\sqrt{2}-1})$?

I am trying to determine if the number $3$ stays prime in the ring of integers of the quartic field $K=\mathbb{Q}(\sqrt{2\sqrt{2}-1})$, or rather adjoin a real root of $X^4+2X^2-7$. I do know ...
1
vote
2answers
87 views

Selfconjugate prime ideal of a cyclic extension of an algebraic number field of prime degree.

Let $K$ be an algebraic number field. Let $A$ be the ring of integers in $K$. Let $L$ be a cyclic extension of $K$ of degree $l$, where $l$ is a prime number. Let $B$ be the ring of integers in $L$. ...
0
votes
0answers
21 views

Ratio of maximum and minimum value

I've tried this so far. Let no. of $-1s$ are $a$, no. of $1s$ are $b$ and no. of $2s$ are $c$. Now $-a+b+2c=19, a+b+4c=99$ On adding and subtracting the equations $a= 40-c$ and $b=59-3c$ Now ...
7
votes
1answer
44 views

Unramified primes of splitting field

I would like to show the following: Theorem: Let $K$ be a number field and and $L$ be the splitting field of a polynomial $f$ over $K$. If $f$ is separable modulo a prime $\lambda$ of $K$, then $L$ ...
1
vote
0answers
26 views

Computing ideal class group by other means than the Minkowski bound?

When calculating the ideal class group of a number field, it is common to start with the Minkowski bound, followed by decomposing finitely many prime ideals of norm less than that bound, and finding ...
2
votes
2answers
38 views

Question about S.Lang's proof of Kummer's Lemma

I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let $K = \mathbf{Q}(\xi_p)$ the $p$-th cyclotomic field extension of $\mathbf{Q}$. Let $u$ be ...
0
votes
1answer
34 views

A diophantine equation of degree 3

Find the integer solutions of $y^2+6=x^3$. I guess it does not have integer solutions but I cannot prove it. By $\pmod 8$, I can know that $y$ is odd and $x\equiv7 \pmod 8$. Then what else can I do?
0
votes
0answers
13 views

Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
2
votes
0answers
25 views

Dirichlet character of order $4$ and the splitting of $p$ in $\mathbb{Z}(\sqrt{-1})$

For $p \equiv 1 \pmod{4}$, let $\psi$ be one of the two Dirichlet characters of order $4$ in $(\mathbb{Z} / p \mathbb{Z}) ^\times$. Consider the character sum $S = \sum_{x=0}^{p-1} \psi(x^2 - a)$, ...
1
vote
1answer
29 views

Subgroup of idele class group is open

On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup $$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$ ...
2
votes
0answers
60 views

Evaluating the norm of an ideal by considering integral basis for $\mathcal{O}_K$

There is this trick in my lecture notes that I don't think is correct. Let $K=\mathbb{Q}(\sqrt{-5})$ then my notes say that the ideal $(2,1+\sqrt{-5})$ in $\mathcal{O}_K$ has obviously norm $2$ since ...
0
votes
1answer
18 views

Are the sets of valuations uniquely determined [on hold]

Let $\mathcal{V_K}$ be the set of valuations of a number field $K$. Can it be that $\mathcal{V_L}=\mathcal{V_K}$, for the set of valuations of another number field $L$ non-isomorphic to $K$?