Questions related to the algebraic structure of algebraic integers
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0answers
31 views
ideals in rings of algebraic integers are finitely generated
I am trying to write about rings of algebraic integers $\mathcal{O}_K$ in a number field $K$ without introducing to much field theory. I want to show that these rings are Dedekind. First of all I want ...
4
votes
1answer
49 views
Integral basis for a number field
I need some help in solving the following problem:
Suppose $K$ is a number field and $K=\mathbb{Q}(\theta)$ where $\theta\in\mathfrak{O}_K$, the ring of integers of $K$. Now among the elements in ...
0
votes
2answers
66 views
Prove that $\sqrt{1+\pi^2}, \hspace{2pt}\pi-\sqrt\pi,\hspace{2pt} \pi^2+\pi+\sqrt{1+2\pi}$ are not algebraic
Considering the fact that $\pi$ is not an algebraic number, I need to prove these expressions are not algebraic :
$$\sqrt{1+\pi^2},\hspace{10pt} \pi-\sqrt\pi,\hspace{10pt} \pi^2+\pi+\sqrt{1+2\pi}$$
...
5
votes
3answers
54 views
Find generator of principal ideal
The ideal $(9, 2 + 2\sqrt{10})$ of $\mathbb{Z}[\sqrt{10}]$ is a principal ideal; it is generated by $1+\sqrt{10}$.
This is easy enough to check once it's been found, but can anyone tell me some way ...
1
vote
1answer
33 views
Do $R'$ integral over $R$ imply $\operatorname{Frac}(R')$ algebraic over $\operatorname{Frac}(R)$?
This is Theorem 6.6 from Janusz, Algebraic Number Fields, it says:
Let $R\subseteq R'$ be Dedekind domains with $R'$ integral over $R$ and $\mathfrak{p}$ a nonzero prime ideal of $R$. Suppose
...
2
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1answer
32 views
How to calculate the norm of an ideal?
Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly.
For example, I know that the norm ...
3
votes
1answer
34 views
Proving convergence of a Hilbert modular theta function $\vartheta(z):= \sum\limits_{x \in \mathcal{O}_F} e^{\pi i \operatorname{Tr}(x^2 z)}$
I'm trying to understand a somewhat sketchy proof that I found online of the convergence of the analog of Jacobi's theta function $\displaystyle{\theta(\tau) := \sum_{n = -\infty}^{\infty} e^{2 \pi i ...
5
votes
1answer
40 views
Extension of valuation to the algebraic extension of a number field.
I am trying to get the idea how we can extend the $p$-adic valuation on $\mathbb Q$ to an algebraic extension. In particular, how to extend the $p$-adic valuation for $p = 5$ from $\mathbb Q$ to ...
4
votes
1answer
56 views
Irreducible ideal implies prime ideal in Dedekind Domains?
An ideal is irreducible if it can not be written as the finite intersection of strictly larger ideals. In a Noetherian ring every irreducible ideal is primary, but the converse doesn't hold. I wonder ...
9
votes
2answers
78 views
Showing that a real number is an algebraic integer
For what values of $x,y,z\in\mathbb{Z}$, such that $0\leq x,y,z\leq 2, $ the real number $$\alpha:=\frac{1}{3}\left(x+\sqrt[3]{175} \cdot y+\sqrt[3]{245}\cdot z\right)$$ is an algebraic integer i.e. ...
2
votes
1answer
42 views
A calculation of the norm of an ideal
Let $L$ be a number field of degree $n$ over $\mathbb{Q}$ and $\mathfrak{a}$ a non-zero ideal of the ring of integers $\mathcal{O}_L$. Suppose that $X=\{x_1,...,x_n\}$ is a $\mathbb{Z}$-basis of ...
2
votes
0answers
56 views
Proof of Stickelberger’s Theorem
I am having some trouble in understanding the proof of Stickelberger’s Theorem,
$\textbf{Theorem :}$ If $K$ is an algebraic number field then $\Delta_K$, the discriminant of $K$, satisfies ...
1
vote
0answers
37 views
Algebraic Manipulation [closed]
An operation produces $A−\frac{1}{A}$ from a fraction $A=\frac{m}{n}$, where $m \ne n$ and $ m \ne 0$. If the initial value of $A$ is $\frac{24}{47}$ and the operation is repeated 2012 times, the ...
4
votes
1answer
63 views
Residue fields of gaussian primes
I just started with algebraic number theory and need some help: In the ring of Gaussian integers a prime $p$ with $p=1\bmod 4$ splits into a product of two irreducible elements $(a+bi)(a-bi)$, with ...
2
votes
2answers
35 views
Prime decomposition in ring extensions
Let $R\subseteq R'$ be Dedekind domains, let $\mathfrak{p}$ be a nonzero prime ideal of $R$. Then $\mathfrak{p}R'$ is an ideal of $R'$ and it has a factorization
...
4
votes
2answers
91 views
Class Group of $\mathbb Q(\sqrt{-35})$
As an exercise I am trying to compute the class group of $\mathbb Q(\sqrt{-35})$.
We have $-35\equiv 1$ mod $4$, so the Minkowski bound is $\frac{4}{\pi}\frac12 \sqrt{35}<\frac23\cdot 6=4$. So we ...
1
vote
1answer
37 views
Definition: Gauss Sum - Where is the error?
In my algebraic number theory lecture we defined Gauss sums as follows. However, I am quite unsure whether this definition is correct (our lecturer is quite absentminded at times). My intuition says ...
7
votes
2answers
70 views
Show field of fractions is finite extension of $\mathbb{Q}$
Let $A$ be a ring which is also a finitely generated $\mathbb{Z}$-module. If $A$ is an integral domain and $K$ is its field of fractions and $K$ has characteristic zero, then why is $K$ a finite ...
6
votes
1answer
134 views
All number fields with absolute value of discriminant $\le 20$
I need to find all number fields with absolute value of discriminant $\le 20$.
Using Minkovsky's theorem I understood that it should be quadric or cubic extension. The case of quadric is very ...
6
votes
4answers
195 views
number of solutions to an equation?
Given $x$ and $y$ are multiples of $2$ satisfying
$$x^2 - y^2 = 27234702932$$
Find the number of solutions to $x$ and $y$.
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votes
0answers
38 views
Help me prove the supremum property.
Let $A$ and $B$ be nonempty sets and $f$ be a function from any nonempty set $S$ to subset of real number. Prove that
$$\sup_{x \in A} \{ \min \{ \sup_{y \in B} \{ \min \{ f(y) \} \}, f(x) \} \}= ...
1
vote
1answer
36 views
extended Euclidean (xgcd) in quadratic integer rings
Given a discriminant $D < 0$, I have the quadratic imaginary field $\mathbb{K} := \mathbb{Q}(\sqrt{D})$. And the quadratic integer ring is given by
$\mathcal{O} = \mathbb{Z} + \mathbb{Z} \frac{D + ...
1
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0answers
26 views
The criteria for two abelian extensions to be embedded
Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help.
Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
6
votes
3answers
128 views
Preparations for reading Algebraic Number Theory by Serge Lang
I am eager to learn algebraic number theory. It seems that Serge Lang's Algebraic Number Theory is one of the standard introductory texts (correct me if this is an inaccurate assessment). I flipped ...
3
votes
1answer
33 views
If $L\mid K$ is finite not separable then the pairing $\langle x,y\rangle =T_{L\mid K}(xy)$ is degenerate
Let $L$ over $K$ be a finite not separable extension of fields. I want to prove that
$$\langle x,y\rangle:=T_{L\mid K}(xy)$$
is degenerate, i.e. there exists a nonzero $x\in L$ such that $\langle x, ...
4
votes
2answers
85 views
How to prove an algebraic integer.
How can you prove that something is an algebraic integer? I have two examples:
1) $\frac{\sqrt{p} + \sqrt{q}}{2}$ , where $p$ and $q$ are integers congruent to $3 \pmod 4$
2) $\sqrt{\frac{\sqrt{15} ...
11
votes
1answer
93 views
Introduction to the trace formula for people outside number theory
I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula.
I am aware that there are "standard references" on the ...
32
votes
3answers
678 views
What are examples of unexpected algebraic numbers of high degree occured in some math problems?
Recently I asked a question about a possible transcendence of the number ...
4
votes
0answers
54 views
Galois representations and normal bases
I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a ...
3
votes
2answers
42 views
Let $p$ be an odd prime. $p$ and $(1-\zeta_p)^{p-1}$ are associates in $\mathbb{Z}[\zeta_p]$.
Let $p$ be an odd prime and $\zeta_p$ a primitive $p$th root of unity, that is a $p$th root of unity other than 1. I need to show that $p$ and $(1-\zeta_p)^{p-1}$ are associates in ...
3
votes
1answer
45 views
A proof in Janusz Algebraic Number Field
I can't understand Lemma 6.2 from the Janusz book Algebraic number fields, pag. 26, that says:
Let $A\subset B$ be integral domains with $A$ integrally closed and $B$ integral over $A$. If ...
1
vote
1answer
34 views
fraction field of the integral closure
Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of ...
10
votes
3answers
139 views
What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?
I am currently doing a past paper and it asks the following:
Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements.
If I assume standard algebraic number ...
1
vote
0answers
21 views
Additive characters of Quotient ring
Suppose $K$ is finite Galois extension of rationals, and let $\mathcal O$ be the ring of integers of $K$,
and finally suppose $\mathfrak m$ is an integral ideal of $\mathcal O$,
I want to classify ...
3
votes
1answer
44 views
Express in terms of familiar arithmetic functions
How can I express the sumation $$h_k(n)=\sum_{d|n, k|d}\mu (d)$$ in terms of familiar arithmetic functions, where $k\in \mathbb{N} $ is fixed?
1
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0answers
32 views
Is it always possible to find primes $p, q$ such that $\left(\dfrac{p}{q}\right)_n=\left(n,\dfrac{p-1}{f}\right)=1$?
I first provide a background, or the context, where this question arises. Skip it if one wants so.
Background:
In the book The Genus fields of algebraic number fields by Ishida, one finds the ...
5
votes
1answer
66 views
Showing that $\mathbb Q(\sqrt{17})$ has class number 1
Let $K=\mathbb Q(\sqrt{d})$ with $d=17$. The Minkowski-Bound is $\frac{1}{2}\sqrt{17}<\frac{1}{2}\frac{9}{2}=2.25<3$.
The ideal $(2)$ splits, since $d\equiv 1$ mod $8$. So we get ...
6
votes
2answers
137 views
Solve: $x^2-py^2=q$
Solve $$x^2-py^2=q$$ for integers $x,y$, here $p,q$ are both given prime numbers.
It's obvious that $p,q$ should satisfy $(\frac{p}{q})=(\frac{q}{p})=1,$ here $(\frac{p}{q})$ is the Jacobi symbol.
...
5
votes
2answers
142 views
Are algebraic numbers analogous to group elements with finite order?
Would you say that the "elements with finite order" in group theory are analogous to "algebraic numbers" in field theory?
I thought this is the case since requiring an algebraic number $\alpha$ to be ...
2
votes
1answer
41 views
fractional ideals in the localization of a Dedekind
I'm reading Janusz, Algebraic number fields, 1973, pag.16-17, where defining a fractional ideal of a Dedekind domain $R$. A fractional ideal of $R$ is a non-zero finitely generated $R$-submodule ...
5
votes
2answers
72 views
Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$
I'm studying for my algebra quals this fall and keep encountering problems like the following:
List all the ideals of $\mathbb{Z}[x]/(16, x^3)$.
or
List all the primes of ...
1
vote
0answers
34 views
Endomorphisms of the multiplicative formal group law
Is there a simple description of the ring of endomorphisms $\mathrm{End}(\mathbb{G}_m)$ of the formal group law $$\mathbb{G}_m(X,Y) = X + Y + XY,$$ at least over a ring of characteristic zero? I'm ...
1
vote
3answers
66 views
Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$
For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$
where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
2
votes
2answers
27 views
Is the inverse of a fractional ideal still fractional?
Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define
...
0
votes
0answers
32 views
$\langle v,\sqrt{2}v\rangle_{\mathbb{Z}}$ not a discrete subgroup of $\mathbb{R}^{2}$ [duplicate]
I got a list of exercises to do and there is one of the first exercises which I do not manage to solve.
Its statement is the following:
Let $v\in \mathbb{R}^{n}$ be a nonzero vector. Using the fact ...
3
votes
3answers
110 views
Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers
Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal?
My notes say:
$1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
8
votes
3answers
188 views
Solve $x^3+x \equiv 1 \pmod p$
Find all the primes $p$ so that $$x^3+x \equiv 1 \pmod p \tag{1}$$ has integer solutions.
We consider $x$ and $y$ are the same solutions iff $x\equiv y \pmod p.$
We can prove that $(1)$ cannot ...
3
votes
1answer
40 views
How to determine a Hilbert class field?
I tried to solve the exercise VIII.XX in Number Fields by Marcus. It asks to find the Hilbert class field of $Q(\sqrt m)$ for $m=-6,-10,-21,-30$. And the emphasis of this question is on the first two. ...
1
vote
2answers
40 views
Nonreal units in totally imaginary number fields
Suppose we are given a totally imaginary number field $L$ of degree greater $2$, is it possible that all units of $L$ lie in $\mathbb R$? This is true for almost all quadratic imaginary number fields, ...
5
votes
1answer
106 views
Vague definitions of ramified, split and inert in a quadratic field
Our lecturer defined the following:
Let $K=\mathbb Q(\sqrt d)$ be a quadratic field and $p$ a prime number, then
(1) $\ p$ is ramified in $K$ if $\mathcal O_K⁄(p)\cong \mathbb F_p [x]⁄(x^2)$
...
