Questions related to the algebraic structure of algebraic integers

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5
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0answers
68 views

Show that $\sum_{d\mid f} \varphi(f/d) a^{|d|} \equiv 0 \pmod f$

This equation is correct when $f$ and $a$ are any integers. I want to show that this holds for $f,a\in K[x]$ where $K$ is any finite field. In the equation $\varphi(f)$ is defined as $|(K[x]/(f))^\...
6
votes
2answers
73 views

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
3
votes
0answers
44 views

Finding units in quadratic integer rings

I want to find the units in $\mathbb{Z}[\alpha]$, where $\alpha=\frac{1+\sqrt{-11}}{2}$. One can of course use norms to find the units in quadratic integer rings of the form $\mathbb{Z}[\sqrt{D}]$ ...
2
votes
2answers
45 views

How to find the class number of $\mathbb{Q}(\sqrt{-17})$?

I tried to calculate the class number with help of the Minkowski bound of $M \approx 5$. So if an ideal has norm $1$, it is the ring of integers. If it has norm $2$, it is $(2, 1+\sqrt{-17})$, which ...
0
votes
0answers
23 views

Embeddings of $K_v$ in $\mathbb{C}$

Let $K$ be a number field, $v$ a nonarchimedean prime, and $K_v$ the completion of $K$ at $v$. We have the embedding $K \to K_v$, and also $K \to \mathbb{C}$. I have two related questions: Is ...
1
vote
1answer
36 views

construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$?

how might I construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$? Can it be done using pigeonhole principle as with square roots and Pell equation. I had been reading about the Voronoi continued fraction or ...
1
vote
1answer
30 views

Understanding a proof in Washington's “Cyclotomic Fields”

I'm working through Washington's "Cyclotomic Fields" and having a problem with the proof of Proposition 3.8, which states: Given an abelian group G, there is an everywhere-unramified extension of ...
2
votes
1answer
129 views

What is number theory today? [closed]

I try to explaine my problem and I hope do not disturb or annoy; I know that number theory is very vast but essentially it is divided into two parts: analytic number theory and algebraic number ...
1
vote
2answers
69 views

Galois principle for ideals

Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Determine a necessary and sufficient condition on $L/K$ to ensure that $$\{I\in \text{Id}_L,\text{ such that }\sigma (I)...
2
votes
4answers
158 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
3
votes
0answers
54 views

How a complex root $\eta$ of $x^2 + x + A$ affects the ring $\mathbb{Z}[\eta]$

While reading a statement in P. Pollack's Not Always Buried Deep: A Second Course in Elementary Number Theory I came across a statement that seemed obvious and I am wondering if I am oversimplifying ...
1
vote
2answers
60 views

Restriction from subgroup of the Galois group of max. unr, ext. $G(\tilde{K}/\mathbb{Q}_{p})$ to $G(K/\mathbb{Q}_{p})$ is surjective?

This is a question I'm struggling with for some time. Let $K$ be a finite Galois extension of $\mathbb{Q}_{p}$ and let $\tilde{K}$ denote the maximal unramified extension of $K$. We can then ...
1
vote
0answers
36 views

Inertia of an elliptic curve with potentially good reduction

Let $E/\mathbb{Q}_p$, $p\geq5$ be an elliptic curve with additive potentially good reduction. Then there is a unique, minimal, finite and totally ramified extension $K$ such that $E/K$ has good ...
2
votes
2answers
44 views

Finding the degree of $\sqrt[3]{2} + \sqrt[3]{3}$ over $\mathbb Q$ [duplicate]

I am practicing writing down random algebraic numbers and finding their degrees over $\mathbb Q$ and have fumbled when coming to $\sqrt[3]{2} + \sqrt[3]{3}$. Mathematica tells me the minimal ...
4
votes
2answers
69 views

Galois group of $\overline{\mathbb{F}_{p}}$ gives arithmetical information for finite fields $K/\mathbb{F}_{p}$?

Let $\mathbb{F}_{p}$ be the field with $p$ elements and $\overline{\mathbb{F}_{p}}$ be its algebraic closure. For some reason, we want to understand the structure of the Galois group of such an ...
3
votes
0answers
43 views

Image of the norm map in imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^...
1
vote
0answers
26 views

$p$ ramify in $\mathcal{O}_K$ means repeated root modulo $p$?

I read that if $p$ ramifies in $K$ the splitting field of a polynomial $f$ on $\mathbb{Q}$, then $p\mathcal{O}_K$ has a repeated factor. How does this lead to that $f$ modulo $p$ has a repeated root?
1
vote
0answers
35 views

Explicit degree two Artin L-function

Let $K$ be a cubic field, and $\zeta_K(s)$ the Dedekind zeta function. Then from here one has the factorization $$\frac{\zeta_K(s)}{\zeta_{\bf Q}(s)}=\sum_{n=1}^\infty\frac{a(n)}{n^s}$$ where $a(n)=\...
1
vote
1answer
38 views

Nice shapes of ideals of $\mathbb{Z}[i]$ from a (lattice) geometric point of view?

If we draw the lattice for the ideal generated by $(2+i)$ in $\mathbb{Z}[i]$, and look at what is happening modulo $(2+i)$, we see a beautiful square, although it is rotated a little bit counterclock-...
6
votes
2answers
59 views

How to construct rings with a given class number?

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.
3
votes
2answers
93 views

Does the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$ have a power basis?

Specifically I am interested in the the ring of integers for the field $\mathbb{Q}(\sqrt{-1+2\sqrt{2}})$. Does this ring of integers have a power basis? More generally, for any Salem number $s$, ...
2
votes
1answer
46 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 ...
4
votes
1answer
46 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
32 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 $(2)$ $g=\sum_{t} \zeta^{t^2}$ ...
5
votes
1answer
99 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 $\mathrm{Gal}(\...
0
votes
1answer
45 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
1answer
57 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
0answers
56 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 $\exists b \in I$ ...
4
votes
1answer
151 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 ...
0
votes
0answers
51 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
50 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 : \mathbf{Z}_p^\times(1+...
0
votes
0answers
52 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 $q=(...
2
votes
1answer
54 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 ...
0
votes
0answers
42 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 $...
2
votes
0answers
89 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 ...
4
votes
1answer
43 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 ...
1
vote
1answer
17 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 ...
5
votes
2answers
87 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 ...
1
vote
2answers
32 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
60 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
0answers
38 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 ...
3
votes
1answer
122 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 $N$ (i.e, ...
2
votes
1answer
72 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?
9
votes
1answer
107 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 $\mathbb{Z}[\...
10
votes
2answers
249 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
86 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
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
44 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 ...
1
vote
1answer
57 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 Q[$\sqrt{2}+\sqrt{3}$]=Q[...