The algebraic-number-theory tag has no wiki summary.
-5
votes
0answers
121 views
Generalization of prime numbers [on hold]
I transcribe the text from my unpublished work and it may be new contribution over the field of prime number theory. In particular, I shall not expand it to up - because I am not acquired with math ...
2
votes
1answer
137 views
Modular Functions with Rational Fourier Expansions
I have been reading the paper of Cox, McKay and Stevenhagen "Principal Moduli and Class Fields", http://arxiv.org/pdf/math/0311202v1.pdf, and I have a question regarding the nature of the function ...
2
votes
2answers
131 views
What is the exact meaning of the real period in the $p$-adic formulation of BSD?
Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism ...
23
votes
6answers
742 views
Patterns among integer-distance points
Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...
2
votes
1answer
152 views
Prime ideals in the ring of algebraic integers
Let $m(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1 x+ a_0$, $a_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}(x) / {m(x)\mathbb{Q}(x)}$. K is an algebraic number ...
1
vote
0answers
31 views
Degrees of the real and imaginary parts of an algebraic number [migrated]
(1) If x=a+bi is a root of an irreducible polynomial over the rationals of degree n, what is the maximum possible degree of the irreducible polynomials over the rationals for the real numbers a and b?
...
3
votes
0answers
213 views
Is the infimum of Salem numbers > 1?
BACKGROUND
A Salem number is an algebraic integer $\theta$ such that all the Galois conjugates of $\theta$ are $\leq 1$ in absolute value, and at least one of them lies on the unit circle. Their ...
0
votes
1answer
325 views
How to do such a partitioning?
Assume:
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N}
$$
and,
$$
f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l}
$$
I am going to ...
6
votes
2answers
239 views
Class numbers of orders
Consider an order $R$ in a number field $L$. Let $C_R$ be the set of $R$-fractional ideals modulo $L^\times$. Let $O$ be the maximal order in $L$, and $C_O$ be the class group of $O$.
My question: ...
1
vote
0answers
80 views
Reference request for a basic result on relative differents & discriminants
I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), ...
7
votes
1answer
140 views
Class number of real maximal subfield of cyclotomic fields
Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$?
Are there infinitely many? Finitely many? ...
8
votes
1answer
247 views
Fundamental units of imaginary quartic fields
Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...
3
votes
1answer
158 views
$\ell$-conductor of a two-dimensional $\ell$-adic Galois representation
Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...
0
votes
0answers
52 views
Divisor bounds of ideals in number fields
Let $K$ be an algebraic number field and let $I$ be an ideal in $O_K$ (the ring of integers).
Denote by $d(I)$ the number of ideals that divide $I$.
So if $I= \prod_{i=1}^k p_i^{e_i}$ is the ...
3
votes
0answers
67 views
How to estimate a local hilbert samuel funcion
Let $X$ be a reduced hypersurface in the projective variety $\mathbb{P}^n(K)$, where $K$ is a number field. Select $\xi$ is a $F_{\mathfrak{p}}$-rational point of $X$ where $\mathfrak{p}$ is a prime ...
4
votes
1answer
279 views
Analogy between Jacobian of curve and Ideal class group
It is excerpt from "Algebraic Geometry Codes Basic ...
-1
votes
1answer
190 views
Why do no prime ideals ramify in the extension $\mathbb{Q}(\sqrt{p }, \sqrt{q})/\mathbb{Q}(\sqrt{pq })$? [closed]
Let $p,q $ be odd integer primes, $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$.
$K = \mathbb{Q }[\sqrt{pq }]$, $L = \mathbb{Q}[\sqrt{p }, \sqrt{q} ]$. Why a prime ideal in $O_{K}$ doesn't ramify in ...
1
vote
2answers
109 views
On Cubic Non-Residues Modulo a Prime [closed]
What is a good test for identifying cubic non-residues/residues and higher power non-residues/residues modulo a prime $R$ in terms of computational complexity?
Given $M$ and $N$, is there a good way ...
2
votes
0answers
79 views
What is the real subring of a ring of cyclotomic integers?
I am looking at tilings whose vertices lie in a ring of cyclotomic integers. These tilings are of interest as they can have interesting scaling properties or be substitution tilings. Interesting ...
7
votes
1answer
525 views
Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?
From a naive outsider's viewpoint, just watching the MO postings
in those three fields scroll by, and hearing of breakthroughs in the news,
it appears there might be increasing overlap among the ...
15
votes
1answer
389 views
Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?
By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$?
Thank you!
0
votes
1answer
162 views
For any n and some prime p there is an elemnet in Zp* of order n [closed]
How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...
1
vote
1answer
134 views
Lower Degree Elements in an Algebraic Number Field
Fix an algebraic integer $\alpha$ of degree $n$
such that the extension $K=\mathbf{Q}(\alpha)/\mathbf{Q}$ has intermediate fields.
(We can assume $K$ is Galois with non-simple Galois group.)
This ...
2
votes
0answers
90 views
classification of rank $2$ $\mathbb{Z}/p^n\mathbb{Z}$-algebra with invertible discriminant
Let $p$ be a prime number and $n$ be an integer. Let $A$ be an $\mathbb{Z}/p^n\mathbb{Z}$-algebra of rank $2$ whose discriminant is non invertible. In Serre's book lecture on the mordell Weil theorem ...
7
votes
2answers
271 views
Quintic polynomial solution by Jacobi Theta function.
Does someone have a good and rigorous reference for the solution of quintic ploynomial equation with Jacobi Theta function, in English?
Mathworld and Wikipedia don't give a good English reference, at ...
6
votes
0answers
56 views
Is the equidissection spectrum closed under addition?
If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...
8
votes
1answer
279 views
Extensions of Galois representations
Let $G=Gal(\bar{\mathbb Q}/{\mathbb Q})$ be the absolute Galois group of the rationals. Fix two continuous group homomorphisms $\alpha,\beta: G\to {\mathbb Q}_l^\times$, where $l$ is a prime and ...
40
votes
5answers
1k views
If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...
4
votes
1answer
404 views
Doubt in the proof of Stickelberger's Theorem
I was going through the proof of Stickelberger's Theorem, as given in the book 'Algebraic Number Theory' by Richard A Mollin, and I am having some problem in understanding the proof. I will state the ...
2
votes
0answers
397 views
Simplifying an algebraic integer expression
I have an expression where the variables are algebraic integers:
$p4 = \frac{p12 - p41 \cdot p21}{p22}$
p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and p21 is degree 24. I ...
0
votes
0answers
83 views
Bounding number of solutions to an equation:
I have an equation that I think should not have too many solutions, but I don't see a way to argue this.
Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, y \leq N$ can the ...
16
votes
0answers
719 views
Orders in number fields
Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$.
Question: Let $p$ be an ...
2
votes
0answers
64 views
Decompositions of representations of pro-p groups
Let $P$ be a pro-p group. Assume that there is a filtration of $P$ by normal subgroups $P_i$ such that $P_0=P$ and $P_{i+1} < P_i(i\in\mathbb N)$. Let $V$ be an $l$-adic representation of $P$, ...
1
vote
1answer
103 views
ramification of discrete valuation field
Let $K$ be a discrete valuation field with valuation $v:K\rightarrow \mathbb Z\cup \{\infty\}$ which is normalized by $v(\pi)=1$ for a prime element $\pi$. Let $v:\overline K\rightarrow \mathbb ...
0
votes
0answers
67 views
Artin L- Function properties
Hi, 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 ...
1
vote
0answers
66 views
points in $V(\bar K \otimes_{\bar Q} \bar L)$ rational over tensor product of fields
Let V be a variety over a number field, and let K and L be two algebraically closed
What is known about the points of $V(\bar K \otimes_{\bar Q} \bar L )$ ?
Are there results claiming that points in ...
8
votes
1answer
377 views
how to visualize the class number of an imaginary quadratic field?
Let me detail the title of the question. I'm trying to give students an intuition of what the class number is.
Let $K=\mathbb{Q}(\sqrt{-d})$, with $d>0$ a square-free integer, be a quadratic ...
2
votes
0answers
103 views
P-adic Weierstrass Lemma for several variables
The p-adic Weiestrass lemma asserts that a power series $f(z)$ with coefficients in the ring of integers of a local field can be factored as $π^n·u(z)·p(z)$ where u(z) is a unit in the ring of power ...
5
votes
1answer
224 views
Inertia subgroup in the ordinary reduction case when $p=2$
Dear MO,
Let $K/\mathbb{Q}_2$ be a finite extension, and let $E/K$ be an elliptic curve with good ordinary reduction, and such that $\mathbb{Q}_2(j(E))=K$. Let ...
7
votes
1answer
351 views
Numbers integrally represented by a ternary cubic form
Given integers $a,b,c,$ and cubic form
$$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$
$$ f(a,b,c) =
\det \left( \begin{array}{ccc}
a & b ...
4
votes
3answers
253 views
Computing certain class numbers modulo 4
Let $p \equiv 5 \pmod{8}, q \equiv 7 \pmod{8}$ be primes and $N = pq$. I want to show that the class number $n$ of $\mathbb{Q}(\sqrt{-N})$ satisfies $n \equiv 2 \pmod{4}$ if $\left(\frac{q}{p}\right) ...
0
votes
1answer
204 views
local field and number field
Let $K$ be a local field (locally compact topological field) of characteristic zero.
Is it true that $K$ is isomorphic to the completion of a number field
under some valuations?
If yes, then how to ...
13
votes
1answer
543 views
Principal maximal ideals in Z[x]/(F)
Is there some irreducible $F \in \mathbb{Z}[x]$ such that $\mathbb{Z}[x]/(F)$ has no principal maximal ideal? Equivalently, is it possible that the $1$-dimensional integral domain $\mathbb{Z}[x]/(F)$ ...
0
votes
0answers
203 views
Topology of the complex p-adic numbers
Let $\mathbb{Q}_p$, be the usual p-adic numbers, and then take the completion of its algebraic closure to get $\mathbb{C}_p$.
I was wondering if I could get a reference or some explanation as to what ...
1
vote
1answer
316 views
How can we understand Baker's theorem about transcendence ?
We know that for algebraic $e^{it\pi}$, $t$ can not be algebraic irrational by
Baker's theorem, but his proof is analytic; is there some algebraic understanding for such fact? If $t$ is ...
0
votes
1answer
301 views
If e^itπ is algebraic , is $t$ a rational number. [closed]
I have a elementary question:If e^itπ is algebraic , is $t$ a rational number.
I do not know whether it is right
0
votes
0answers
57 views
Lower bound on number of solutions to diophantine equations with all but one linear constraint
Hi,
I want to ask a simple question in diophantine systems. I have tried to search under different headings, but was unable to find a suitable answer to my question.
I have a set of $m$ linear ...
5
votes
1answer
223 views
Rational points on surfaces of general type
The weak Lang conjecture asserts that rational points on a variety of general type defined over $\mathbb{Q}$ are not Zariski dense (same replacing $\mathbb{Q}$ with a number field). This one is proved ...
3
votes
2answers
528 views
Exercise in Milne's CFT notes
On page 156 of Milne's Class field theory notes available online here, he claims that the Hilbert class field of $K = \mathbb Q(\sqrt{-6})$ is the splitting field of $x^2+3$ but I don't believe so.
...
1
vote
1answer
158 views
Functional equations of zeta functions over global fields
The functional equations for Dedekind zeta functions (zeta functions attached to rings of integers in algebraic number fields) come from functional equations of theta functions like $\sum_{n \in ...
