Questions related to the algebraic structure of algebraic integers
4
votes
2answers
74 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.
...
4
votes
2answers
108 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
33 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
67 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
32 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
54 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
21 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
31 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
68 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
171 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
34 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
36 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
88 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 $\mathcal O_Kâ(p)\cong \mathbb F_p [x]â(x^2)$
...
2
votes
1answer
37 views
Euclidean algorithm in lattices in $\mathbb{C}$
Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. I want to prove that, for every $\alpha,\beta\in R,\beta\neq 0$, there exist $\gamma,\delta\in R$ such that $\alpha=\gamma\beta+\delta$, with ...
2
votes
1answer
25 views
How to find $\sup(\{|x-y|_p : x,y\in B(0;r)\})$
Just to clarify the notation and the question:
Working in p-adic space $\mathbb{Q}_p$, we have the norm $|x|_p=p^{-ord_p(x)}$ and we define the metric over this space as $d(x,y)=|x-y|_p$. We are ...
1
vote
1answer
25 views
If $R$ is integral over $S$, then $frac(R)/frac(S)$ is finite extension of fields
How to show that:
If $R\supset S$ are rings, $R$ is integral over $S$, $K$ and $L$ the fraction fields of $R$ and $S$ respectively, then $K/L$ is finite extension of fields.
2
votes
1answer
51 views
$\mathbb{Q}[\sqrt a_1,\ldots,\sqrt a_k]$ vs. $\mathbb{Q}(\sqrt a_1,\ldots,\sqrt a_k)$
Call an algebraic number polyquadratic if it can be expressed as the sum or difference of a finite number of square roots of rational numbers. (This definition follows Conway-Radin-Sadun rather than ...
0
votes
0answers
14 views
How to measure the failure of Hasse norm theorem?
We know that the failure the unique factorisation is measured by the ideal class-group, that of the local-global principle depends upon the Tate-Shafarevich group.
Then I thought: what should be ...
3
votes
0answers
33 views
Finiteness of ideal of given norm
I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$.
I know there are "standard proofs" (eg How many elements ...
3
votes
2answers
51 views
Is it true that $\mathbb{Z}_{(p)}=\mathbb{Z}_{p}\cap \mathbb{Q}$?
I know $\mathbb{Z}_{(p)}\subset \mathbb{Z}_{p}\cap \mathbb{Q}$, where $\mathbb{Z}_{(p)}$ is the localization of $\mathbb{Z}$ at prime ideal $(p)$ and $\mathbb{Z}_p$ is the set of p-adic integers. I ...
4
votes
1answer
41 views
On Selmer's curve
I am trying to prove that the equation $3x^3+4y^3+5z^3 \equiv 0 \pmod{p}$ has non-trivial solutions for all primes $p$. I divide it into 3 cases: $p \equiv 0,1,2 \pmod{3}$. The cases $p \equiv 0,2 ...
3
votes
0answers
40 views
Intuition for Krasner's Lemma
From Milne's Algebraic Number Theory, we have (he assumes that $K$ is complete with respect to a discrete nonarchimedian absolute value, but I don't know where the discrete part is being used)
Let ...
1
vote
0answers
24 views
Artin L- Function properties
I'm trying to understand the proof of one of the properties of the Artin L-function. I have the following doubts;
Why take on $f_i =|G_{P_i}: H_{P_i}I_{G,P_i}|$, $H_{P_i}I_{G,P_i}$? and not only ...
4
votes
3answers
71 views
Proof of Hasse-Minkowski over Number Field
Can anyone point me towards a good reference which contains the proof of the Hasse-Minkowski theorem of quadratic forms over a number field? Serre's "A Course in Arithmetic" has a self contained proof ...
1
vote
0answers
66 views
Solve the equation $x^4+y^4=d*z^2$
Solve the equation:$$x^4+y^4=d*z^2,$$
where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer.
I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
3
votes
1answer
58 views
p-adic modular form example
In Serre's paper on $p$-adic modular forms, he gives the example (in the case of $p = 2,3,5$) of $\frac{1}{Q}$ and $\frac{1}{j}$ as $p$-adic modular forms, where $Q = E_4 = 1 + 540\sum ...
3
votes
1answer
51 views
Norm of ideals in quadratic number fields
I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
1
vote
0answers
51 views
Computing the class number of $\mathbb{Q}(\sqrt{1533157})$
I am trying to compute the class number of $\mathbb{Q}(\sqrt{1533157})$ in Magma. Can anyone explain why it's taking so long to compute? I'm currently running Magma V2.18-7. Below is my code:
...
2
votes
1answer
41 views
Definition of nebentypus in $L$-functions.
In Iwaniec and Kowalski, the term nebentypus is mentioned several times in the book. Every time it seems to just refer to a character $\chi$. Since I don't see the authors defining nebentypus, can ...
0
votes
0answers
49 views
Cyclotomic euclidean number fields
I´m here because I want to repeat my question about the norm-euclidean algorithm in a particluar cyclotomic integer ring.
Let $L=\mathbb{Q}(\zeta_{32})$ and $A=\mathbb{Z}[\zeta_{32}]$. On the page ...
2
votes
1answer
47 views
Whether a domain is Dedekind or not
We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
0
votes
0answers
33 views
Find the number of solutions to the equation [duplicate]
Find the number of solutions to the equation
$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=1$
where $a, b, c, d$ are positive integers and $aâ¤bâ¤câ¤d$.
4
votes
3answers
130 views
Show that there exists $f â \mathbb{Z}$ such that $f^2 + f +1 âĦ 0 \pmod p$.
Let $p âĦ 1 \pmod 3$ be a prime. Show that there exists $f \in \mathbb{Z}$ such that $f^2 + f +1 \equiv 0 \pmod p$.
I know the first few primes of this form are: $7,13,19$
So for example $p=7$ we ...
5
votes
5answers
168 views
Proving $\sqrt{2}\in\mathbb{Q_7}$?
Why does Hensel's lemma imply that $\sqrt{2}\in\mathbb{Q_7}$?
I understand Hensel's lemma, namely:
Let $f(x)$ be a polynomial with integer coefficients, and let $m$, $k$ be positive integers ...
4
votes
1answer
119 views
Fermat's Last Theorem in multiple variables
I was wondering if there was anything we could say about when, given $m$,
$\exists n (\forall x_1,\dots,x_m \in \mathbb{N} ( x_1^n + x_2^n + \dots + x_{m-1}^n \neq x_m^n))$
Fermat's Last Theorem ...
0
votes
0answers
30 views
Proving an ideal is principal
Let $R,\mathfrak{P},\overline{\mathfrak{P}}$ and $p$ be as in this question. I have proved that $\mathfrak{P}\cdot\overline{\mathfrak{P}}=pR$. I think this can be used for proving what follows, by I ...
3
votes
1answer
44 views
Proving a factorization of ideals in a Dedekind Domain
Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
3
votes
1answer
62 views
Why does this class group seem inconsistent?
I'm computing a class group for an imaginary quadratic field and something seems wrong.
Let $\delta=\sqrt{-29}$ and let $R$ be the ring of integers in $\mathbb Q[\delta]$. From ...
5
votes
2answers
102 views
Ideals in a Dedekind domain localized at a prime ideal
Let $R$ be a Dedeking domain, let $\mathfrak{i}$ be a non-zero ideal of $R$. By factorization theorem we can write
$$\mathfrak{i}=\mathfrak{p}_1^{a_1}\cdots\mathfrak{p}_n^{a_n}$$
for distinct non-zero ...
1
vote
2answers
93 views
How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory
I learned in my Intro Algebraic Number Theory class that there exist infinitely many integer pairs $(x,y)$ that satisfy the hyperbola $x^2-ny^2=1$; just consider that there are infinitely many units ...
2
votes
1answer
41 views
Stable points and the fundamental domain of the modular group
Let $\mathbb{\Gamma} = \mathrm{SL_2}(\mathbb{Z})$ be the modular group, $\mathcal{F} = \{z \in \mathbb{C} ;\; \lvert z \rvert \geq 1,\; \lvert \Re (z) \rvert \leq 1/2\}$ its fundamental domain.
How ...
2
votes
1answer
93 views
If $x^p +y^p = z^p$ and $xyz \neq 0$, then $p$ divides $x$ or $y$ or $z$?
I am working on an exercise: If $x^5 +y^5 = z^5$ and $xyz \neq 0$, then $5$ divides at least one of $x$, $y$ or $z$.
I am thinking that the answer involves an application of Kummer's theorem, but I'm ...
7
votes
1answer
73 views
Implications between $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ and $(\frak{a})$ + $(\frak{b})$= $(1)$
In a general commutative ring, $(\frak{a} \cap \frak{b})$ = $\frak{a} \frak{b}$ does not imply ($\frak{a}$) + ($\frak{b}$) = ($1$); whereas ($\frak{a}$) + ($\frak{b}$) = ($1$) does imply $(\frak{a} ...
3
votes
2answers
74 views
Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$
I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$.
I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
4
votes
1answer
127 views
Solving $x^2+19=y^5$
I was given several exercises and there is a particular one, I am not able to solve.
Let it be given that $Pic(\mathbb{Z}[\sqrt{â19}])$ is a finite group of order $3$. Use this to find all integral ...
7
votes
3answers
91 views
Does $\mathbb{F}_p((X))$ has only finitely many extension of a given degree?
We know that $\mathbb{Q}_p$ has only finitely many extensions of a given degree in its algebraic closure. Is it the same for $\mathbb{F}_p((X))$?
7
votes
1answer
99 views
Show the two fields are not isomorphic
Let $p,q,r$ be prime integers with $q\neq r$. Let $\sqrt[p]q$ denote any root of $x^p-q$ and $\sqrt[p]r$ denote any root of $x^p-r$. Please show that $\mathbb{Q}(\sqrt[p]q)\neq\mathbb{Q}(\sqrt[p]r)$.
...
2
votes
2answers
97 views
Roots of unity in $\mathbb{Q}(\zeta_p)$ for $p$ an odd prime
I am confused about a last step in a proof that the only roots of unity in $\mathbb{Q}(\zeta_p)$ are $\pm\zeta_p^j$, where $\zeta_p=e^{2 \pi i/p}$, $p$ is an odd prime and $1\leq j\leq p-1$.
So far ...
2
votes
1answer
46 views
A necessary and sufficient condition for a full lattice over an integral domain
I'm learning about lattices over integral domains and I would be grateful if someone could clarify the following for me.
Let $R$ be an integral domain with quotient field $K$ where $K\neq R$. Suppose ...
6
votes
2answers
87 views
Usage of algebraic geometry in understanding the total Galois group of the rational
A professor of mine remarked in class that: "The tools via which we understand the total Galois group of the rationals comes from algebraic geometry".
Could anyone shed some light on this remark, or ...
