Questions related to the algebraic structure of algebraic integers

learn more… | top users | synonyms

1
vote
0answers
65 views

Probability that the discriminant of a quadratic field is divisible by a given prime number $p.$

Find the probability that the discriminant $D$ of a quadratic field $\mathbb{Q}(\sqrt{d})$ is divisible by a given prime number $p$ (beware: the result is not what you may expect.) This is an ...
12
votes
9answers
216 views

$p = x^2 + xy + y^2$ if and only if $p \equiv 1 \text{ mod }3$?

For a prime number $p \neq 3$, do we have that$$p = x^2 + xy + y^2$$for some $x$, $y \in \mathbb{Z}$ if and only if$$p \equiv 1 \text{ mod }3?$$I suspect this is true from looking at the example$$7 = ...
3
votes
2answers
58 views

What does $p$-integral mean?

I'm currently studying Washington's Introduction to Cyclotomic fields and in Theorem 5.10 I came across the term $p$-integral. What does this mean? To give a bit of context: Let $n$ be even and ...
2
votes
0answers
58 views

Siegel's article “The volume of the fundamental domain for some infinite groups”: trouble with understanding computations

This is the article I mentioned. While the idea of what Siegel is doing in order to compute the volume of the fundamental domain described in the article (the very first one, for there are discussed ...
0
votes
0answers
22 views

Abstract CFT. What does Neukirch mean with exponent here?

In Neukirch's Algebraic Number Theory, there is the following Proposition: If we take $K$ a field with characteristic $5$, $n=4$, and choose $\Delta=K^{\ast\:4}$, wouldn't it imply that the trivial ...
0
votes
1answer
32 views

Galois group of a subextension is a subgroup of the Galois group of the extension?

Let $\overline{\mathbb{F}}_{5}$ be the separable closure of $\mathbb{F}_{5}$ and let $G=G(\overline{\mathbb{F}}_{5}/\mathbb{F}_{5})$ be its Galois group. Say we pick a finite subextension of $\...
2
votes
1answer
78 views

What's the sense behind that lemma?

Please if someone can help and can take 3 minutes I would be so so unbelievably happy because it is really important to me... Thank you :) We assume we have a $m$-th root of unity $\zeta_m=e^{\frac{2\...
2
votes
1answer
35 views

Ramification of extension composition

$L, E\supset K$ are number fields. $L/K$ is normal. And field $M=LE$. Assume $\Omega$ is a prime ideal of M and its intersections with $L, E, K$ are $\mathfrak B,\mathfrak q,\mathfrak p$. $(1)$ ...
2
votes
1answer
61 views

Proof of Kummer's Lemma in S. Langs 'Cyclotomic fields'

I was going through the proof of Kummer's Lemma (stated below) as done in Serge Langs Cyclotomic fields on page 312. Now the author states that by class field theory it suffices to show that $\...
2
votes
3answers
44 views

Extension of finite fields under norm map

If $L/K$ is a finite Galois extension with group $G$, we can define the norm of an element $a\in L$ as \begin{equation} N_{L/K}(a)=\prod_{\sigma\in G}\sigma(a). \end{equation} And obviously, this ...
4
votes
0answers
36 views

Given only the minimal polynomial of a field and approx primitive element, find subfields

I want to solve the following problem. Given: The minimal polynomial of a number field, and a decimal approximation of the field's primitive element. Compute: Is the number field an imaginary ...
2
votes
1answer
46 views

When do imaginary quadratic extensions have unique complex places?

This question has been edited in light of some helpful comments from @AdamHughes below. Let $F$ be a totally real number field, i.e. $F=\mathbb{Q}(t)/m(t)$ where $m(t)\in\mathbb{Z}[t]$ and all roots ...
3
votes
1answer
30 views

Decomposition and inertial fields of primes in $\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$

I recently ran into this old number theory prelim problem. Let $K=\mathbf{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$ and let $\mathcal{O}_K$ be the ring of integers of $K$. Find the ramification index and ...
1
vote
1answer
43 views

Problem from the book Number Fields by Marcus

I have been stuck on the 14(c)th problem of the 3rd chapter from Marcus' Number Fields. Let $K$ and $L$ be number fields, $K \subset L$, $R = \mathbb{A}\cap K$, $S = \mathbb{A} \cap L$. Moreover ...
7
votes
3answers
194 views

Does $p = x^2 + 9y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1 \text{ mod }12$?

For a prime number $p \neq 2$, $3$, does $p = x^2 + 9y^2$ for some $x$, $y \in \mathbb{Z}$ if and only if $p \equiv 1 \text{ mod }12$? A case where this is true as to suggest plausibility: $13 = 2^2 +...
0
votes
1answer
23 views

Taking $d$-th root of an element in the algebraic closure of $\mathbb{Q}$

Suppose I have $\alpha \in \overline{\mathbb{Q}}$, the algebraic closure of $\mathbb{Q}$. Suppose I was interested in an element $\beta$ such that $\beta^d = \alpha$. Does there always exist $\beta \...
3
votes
0answers
55 views

Infinite primes (places) of a number field geometrically

Given a (global) number field $K$, thinking of the affine scheme $\mathrm{Spec}\mathcal{O}_K$ can gige an insight into (at least) some kf the number-theoretic terminology, e.g. ramification or local ...
3
votes
1answer
69 views

Completions of number fields

I would like to prove a statement about completions of number fields, but I'm running into a problem. The statement I want to prove is Let $L/K$ be a Galois extension of number fields, $p$ a prime ...
5
votes
1answer
108 views

A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{12})/\mathbb{Q}(i)$ iff it is generated by $\alpha\in1+3\Bbb{Z}[i]$

Prove that for a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[i]$ which does not divide $3$, $\mathfrak{p}$ decomposes completely in the quadratic extension $\mathbb{Q}(\zeta_{12})/\mathbb{Q}(i)$ ...
10
votes
1answer
286 views

Three angles are linearly independent over $\mathbb{Q}$?

If$$\tan \alpha = 1, \text{ }\tan \beta = {3\over 2}, \text{ }\tan \gamma = 2,$$then does it follow that $\alpha$, $\beta$, $\gamma$ are linearly independent over $\mathbb{Q}$? It is possible to test ...
9
votes
4answers
114 views

Subgroup generated by $1 - \sqrt{2}$, $2 - \sqrt{3}$, $\sqrt{3} - \sqrt{2}$

For a number field $K$, Dirichlet's unit theorem says that$$(O_K)^\times = \mathbb{Z}^{r - 1} \oplus (\text{a finite cyclic group}),$$where $r$ is the number of all infinite places of $K$. An infinite ...
1
vote
1answer
20 views

irreducible in the ring of integers

There is a primitive 12-th root of unity and 5 is not a prime since the minimal polynomial mod 5 is reducible. The problem is I don't know how to show 5 is irreducible or not. What I thought was if ...
2
votes
0answers
53 views

Is $(2, \sqrt{m})$ a principal ideal or not in the ring $\mathbb{Z}[\sqrt{m}]$ [closed]

Let $m$ be a negative even integer. In the ring $\mathbb{Z}[\sqrt{m}]$, is the ideal $(2,\sqrt{m})$ a principal ideal?
3
votes
1answer
54 views

Class number of $\mathbb{Q}(\sqrt{n})$ always even? [closed]

Let $n$ be a negative square-free even integer. Does it necessarily follow that the class number of $\mathbb{Q}(\sqrt{n})$ is even?
5
votes
1answer
50 views

Element of $\mathbb{F}_{p^2}$ of order $p^2$ raised to $p+1$th power is element of $\mathbb{F}_p$?

For any prime number $p$ and any element $a$ of the finite field $\mathbb{F}_{p^2}$ of order $p^2$, do we have$$a^{p+1} \in \mathbb{F}_p \subset \mathbb{F}_{p^2}?$$
2
votes
2answers
67 views

What does algebraic number look like locally?

Is there any theorem characterizing what algebraic number looks like locally (in completion)? For example, do all algebraic numbers live in some $\mathbb{Q}_p$? Does there exist algebraic number in ...
2
votes
1answer
46 views

Why must a zero of $f \in \mathbb{Z}_p[X_1, \dots, X_m] ~ (\text{mod } p^n)$ be simple in order to lift to $\mathbb{Z}_p$?

In chapter II, section 2.2, of J-P. Serre's A Course in Arithmetic, we have the following theorem: Theorem 1: Let $f \in \mathbb{Z}_p[X_1, \dots, X_m]$, $x = (x_i) \in \left( \mathbb{Z}_p \right)^...
3
votes
1answer
92 views

Examples of how to apply algebraic number theory

I am reading about algebraic number theory mainly following milne's notes. But currently I really wonder how such theory can help solve problems of number theory. One example I know is we can use ...
2
votes
1answer
91 views

Is 5 a prime element in the cyclotomic ring of integers?

Given a primitive 12-th root of unity, so its minimal polynomial is $$x^4-x^2+1$$ and hence the degree of its cyclotomic ring of integers is 4. Recently I've learnt about quadratic field and ring of ...
3
votes
1answer
68 views

Examples for abstract class field theory?

I'm starting to get into Abstract Class Field Theory, following Neukirch's famous ANT. The initial setup is basically a profinite group $G$ and a discrete abelian group $A$ on which $G$ acting as ...
4
votes
0answers
100 views

Effect of 'Prime conspiracy' on the fact that prime numbers are the generators of integers [closed]

In Unexpected biases in the distribution of consecutive primes, the authors have discovered that prime numbers have decided preferences about the final digits of the primes that immediately follow ...
6
votes
1answer
125 views

Artin Reciprocity $\implies$ Cubic Reciprocity

I'm trying to understand the proof of cubic reciprocity from Artin reciprocity as outlined in this well-known previous math.SE question and the link KCd mentions there. However, there's one final step ...
4
votes
1answer
74 views

Does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$?

For prime numbers $p$ such that $p \equiv 11$, $13$, $17$, $19 \text{ mod }20$, does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\...
5
votes
1answer
55 views

Genus of extension $\mathbb{C}(T)(\sqrt{T^n + 1})$

Let $k = \mathbb{C}$ and $K$ is the extension $\mathbb{C}(T)(\sqrt{T^n + 1})$ of $\mathbb{C}(T)$ with $n \ge 2$ an even integer. I suspect that the genus of $K$ is $(n - 2)/2$, but all attempts at ...
3
votes
1answer
107 views

Intersection between two integral closures equals an algebraically closed field

Consider an algebraically closed field $k$, a finite field extension $K$ of $k(T)$, the integral closure $A$ of $k[T]$ in $K$, and the integral closure $A'$ of $k[T^{-1}]$ in $K$. Prove that $A \cap A'...
2
votes
1answer
47 views

If $x \in \mathbb{Z}[\alpha]$, for $\alpha$ an algebraic integer, is $x^{-1} N(x) \in \mathbb{Z}[\alpha]$ too?

Let $\alpha \in \mathbb{C}$ be an algebraic integer. Assume $\alpha \notin \mathbb{Z}$ to avoid triviality. So the minimal polynomial of $\alpha$ has the form $$m(x) = x^n + a_{n-1} x^{n-1} + \ldots +...
0
votes
1answer
66 views

Why is the kernel of a Galois representation an open subgroup?

Assume that $E$ is a completion of a number field. Then either $E = \mathbb{R}$ or $\mathbb{C}$, or $E$ is a finite extension of $\mathbb{Q}_l$ for a suitable prime number $l$. If $E = \mathbb{R}$ or $\...
1
vote
0answers
43 views

The module $\Omega^1_{\mathcal{O}_E|\mathbf{Z}}$

I have been learning about Kähler differentials recently. If $E$ is an algebraic number field, then it is natural to consider the $\mathcal{O}_E$-module of Kähler differentials $\Omega^1_{\mathcal{O}...
4
votes
2answers
106 views

Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
0
votes
0answers
40 views

Is a p-adic number field and a finite algebraic extension of it ultrametric?

An ultrametric space is a special kind of metric space in which the triangle inequality is replaced with $d(x,z)\leq\max\left\{d(x,y),d(y,z)\right\}$. Is a p-adic number field and a finite algebraic ...
4
votes
1answer
442 views

If more than one prime number satisfies a given congruence, must an infinite number of primes satisfy that congruence?

I understand that this is kind of a broad question, but if no affirmative proof is known, can anyone give a counterexample?
1
vote
1answer
23 views

Basis of neighbourhoods in a profinite group

The Krull topology in a Galois group $G$ of a Galois extension $L/K$ is defined taking $\sigma\:G(L/M)$, where $M/K$ varies through the Galois finite subextensions of $L/K$, as a fundamental system of ...
3
votes
0answers
36 views

Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
3
votes
0answers
46 views

The hypothetical fibre over an infinite prime in $\mathbb{A}^1_{\mathbb{Z}}$

This is just a speculative question I've been wondering about; I hope that others may find it interesting too, but if it's too vague please let me know! We can picture $\mathbb{A}^1_{\mathbb{Z}} = \...
2
votes
1answer
47 views

The torsion subgroup of principal units $U^{(1)}$

$\newcommand{\U}{U^{(1)}}$ $\newcommand{\O}{\mathcal{O}}$ $\newcommand{\p}{\mathfrak{p}}$ $\DeclareMathOperator{\char}{char}$ $\newcommand{\N}{\mathbb{N}}$ I have a question about the torsion ...
2
votes
1answer
34 views

Some combinations of quadratic residues

Suppose $x\in\{a,b\}$ solve $x^2=m\bmod q$ and $y\in\{c,d\}$ solve $y^2=n\bmod q$ then when do all combinations of $xy\bmod q$ have same least non-negative residue? Supposing we have $w\in\{e,f\}$ ...
0
votes
1answer
36 views

A question about fields of fraction

I know that defining a field of fraction, is a way to extend a ring to a field, and also I know that $\mathbb{Q}$ is a field of fractions of $\mathbb{Z}$. Or Guassian rationals are field of fraction ...
1
vote
1answer
65 views

Resolving the tedious cubic

The equation given to me is $$4x^4 + 16x^3 - 17x^2 - 102x -45 = 0$$ I'm asked to find it's resolvent cubic which is not so difficult to find. But the problem is that the question further asks to find ...
2
votes
0answers
37 views

Describing integral ideals [closed]

Suppose I have a field $K=\mathbb{Q}(\sqrt{-d})$. How does one describe it's integral ideals?
1
vote
1answer
108 views

Factorization problem in cyclic cubic field

Let K/$\mathbb{Q}$ be a cubic number field. Assume that K/Q be Galois with class number 1. Therefore Gal(K/Q) is cyclic cubic group and $\mathcal{O}_K$ is a PID. Let p be a rational prime, p ...