Questions tagged [algebraic-number-theory]
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
208
questions
107
votes
6
answers
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How small can a sum of a few roots of unity be?
Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
- 109k
22
votes
3
answers
2k
views
Hecke equidistribution
For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
- 2,468
20
votes
3
answers
4k
views
sum of squares in ring of integers
Lagrange proved that every positive integer is a sum of 4 squares.
Are there general results like this for rings of integers of number fields? Is this class field theory?
Explicitly, suppose a ...
user2529
39
votes
4
answers
7k
views
Which number fields are monogenic? and related questions
A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus'...
- 6,722
46
votes
2
answers
5k
views
Formal group laws and L-series
Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$
denote its L-function, and set
$$ f(x) = \sum a_n \frac{x^n}{n}. $$
Then Honda has observed that
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$
defines ...
- 32.2k
26
votes
4
answers
3k
views
Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?
How does one prove that the splitting of primes in a non-abelian extension of number fields is not determined by congruence conditions?
user19475
24
votes
2
answers
962
views
Proportion of irreducible polynomials $P$ such that $\mathbf Z[X]/(P)$ is the ring of integers of $\mathbf Q[X]/(P)$
I know that number fields have been the object of many statistical experiments.
Is there some kind of heuristics for the following?
Fix a degree $d$ and fix a bound $N$ on the coefficients of a monic ...
- 2,501
23
votes
3
answers
2k
views
Why are values of Eisenstein $E_2^*$ algebraic integers?
I'm looking for a proof that the following term is an algebraic integer whenever $\tau_N=\frac{N+\sqrt{-N}}{2}$ is a quadratic irrationality with class number $1$:
$$A_N:=\sqrt{-N}\cdot\frac{E_2(\...
- 598
14
votes
2
answers
1k
views
Complex Multiplication and algebraic integers
Let $q=e^{2\pi i\tau}$ and
$$E_2(\tau) = 1 - 24 \sum_{n=1}^\infty\frac{nq^n}{1-q^n}$$ be the Eisenstein Series of weight $2$
and let $E_2^*(\tau) = E_2(\tau) - \frac{3}{\pi\cdot Im(\tau)}$ be the ...
- 598
12
votes
2
answers
3k
views
Eigenvalues of nonnegative integer matrices
Edit
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
What are the possible eigenvalues of nonnegative integer matrices?
Any answer to ...
- 3,309
11
votes
2
answers
1k
views
Positive primes represented by indefinite binary quadratic form
Neil Sloane asked me about commands in computer languages to find the (positive) primes represented by indefinite binary quadratic forms. So I wrote something in C++ that works. This is for the OEIS, ...
- 25.4k
4
votes
2
answers
679
views
On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity
Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
- 14.5k
104
votes
10
answers
17k
views
"Understanding" $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$
I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when ...
- 6,942
59
votes
4
answers
7k
views
Has Fermat's Last Theorem per se been used?
There is a long tradition of mathematicians remarking that FLT in itself is a rather isolated claim, attractive only because of its simplicity. And people often note a great thing about current ...
- 10.8k
49
votes
5
answers
3k
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 no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we ...
- 18.4k
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.2k
45
votes
3
answers
4k
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What was the relative importance of FLT vs. higher reciprocity laws in Kummer's invention of algebraic number theory?
This question is inspired in part by this answer of Bill Dubuque, in which he remarks that the fairly common belief that Kummer was motivated by FLT to develop his theory of cyclotomic number fields ...
- 56.8k
34
votes
4
answers
3k
views
$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
- 10.3k
32
votes
1
answer
4k
views
Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $
EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
- 25.4k
29
votes
1
answer
2k
views
Is pi = log_a(b) for some integers a, b > 1?
Are there integers $a, b > 1$ such that $\pi = \log_a(b)$?
Or equivalently: are there integers $a,b > 1$ such that $a^\pi = b$?
Note that the transcendence of $\pi$ makes this a problem - ...
- 19.5k
28
votes
9
answers
15k
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Suggestions for good books on class field theory
Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...
27
votes
4
answers
5k
views
Avoiding Minkowski's theorem in algebraic number theory.
For any course in algebraic number theory, one must prove the finiteness of class number and also Dirichlet's unit theorem. The standard proof uses Minkowski's theorem. Is there a way to avoid it?
...
- 1,407
27
votes
3
answers
2k
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Where's the best place for an algebraic geometer to learn some algebraic number theory?
There are lots of introductions to number theory out there, but typically they are streamlined to assume as little prerequisite knowledge as possible. I'm looking for a text which does the opposite -- ...
- 61.6k
22
votes
5
answers
2k
views
Local inverse Galois problem
It's a basic fact that a finite Galois extension $L/K$ of a local nonarchimedean field $K$ has solvable (in fact supersolvable [edit: no!]) Galois group $G$. One sees this by using the ramification ...
- 1,052
22
votes
4
answers
1k
views
Hasse principle for rational times square
Does a Hasse principle hold for the property of being a rational times a square ?
Let $a \in \mathbb{K}$ be an element of a number field. Assume that at every place $\mathbb{K}_v$ of $\mathbb{K}$, $a$...
- 474
18
votes
2
answers
3k
views
References for Artin motives
I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
- 7,272
16
votes
6
answers
7k
views
Text for Algebraic Number Theory
I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...
14
votes
1
answer
2k
views
Some questions about the ring Z((x))
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
- 10k
13
votes
3
answers
1k
views
Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module
Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the answer of Daniel Loughran. There is a ...
- 13.6k
13
votes
2
answers
1k
views
Upper bound on answer for Pell equation
A user on MSE, @martin , asked https://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
- 25.4k
12
votes
2
answers
2k
views
What is known about first cohomology of the units in a number field?
Let $K/Q$ be a finite Galois extension with Galois group $G$. Let $U\subset K^\times$ be the group of units. I am interested in any available information about $H^1(G,U)$.
Motivation: in the theory ...
- 3,066
12
votes
3
answers
2k
views
2-torsion in class groups of cubic fields
I was wondering if there are good bounds for the $p$-parts of the class group of a number field $F$ in terms of its discriminant $D_F$. More precisely, the bound for the order of the full class group ...
- 3,349
10
votes
1
answer
393
views
Is $\mathbb{Z}$ universally definable in any number fields other than $\mathbb{Q}$?
In 2009, Jochen Koenigsmann showed that $\mathbb{Z}$ is universally definable in the field $\mathbb{Q}$. My question is, are there any other number fields in which $\mathbb{Z}$ is universally ...
- 4,529
10
votes
2
answers
2k
views
Irreducible polynomials with a root modulo almost all primes
Let $f \in \mathbb{Z}[x]$ be a non-zero polynomial which is irreducible over $\mathbb{Q}$. Suppose that $f$ has a root in $\mathbb{F}_p$ for almost all primes $p$. Must $f$ be linear?
Here by ...
- 20.8k
10
votes
2
answers
5k
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Cohen-Lenstra Heuristics reference
I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the ...
- 1,305
9
votes
3
answers
2k
views
Crystalline Characters
Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}_p$, then $\chi$ must be a ...
- 4,760
8
votes
7
answers
1k
views
Old question of Serre on discriminants of a sequence of polynomials
Let $P_n(t)$ be polynomials with integer coefficients with $d_n = \deg(P_n(t))$ going to infinity when $n$ goes to infinity
and with nonzero discriminants $disc(P_n(t)) \neq 0$.
Question: Is
$$
\...
- 1,601
8
votes
1
answer
926
views
Understanding Umemura's Theorem for roots of algebraic equations
I am trying to understand Umemura's Theorem for expressing the roots of any algebraic equation by higher genus theta functions. The original paper can be found here: Umemura, H.: Resolution of ...
- 483
7
votes
1
answer
794
views
Parity of class number of pure cubic fields
A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.
What can be said about the parity (odd or even) of the class number of a pure ...
- 73
6
votes
1
answer
814
views
Reduction mod $p$ of units in a ring of integers
Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which ...
- 63
6
votes
2
answers
788
views
Is every square root of an integer a linear combination of cosines of $\pi$-rational angles?
For example, $\sqrt 2 = 2 \cos (\pi/4)$, $\sqrt 3 = 2 \cos(\pi/6)$, and $\sqrt 5 = 4 \cos(\pi/5) + 1$. Is it true that any integer's square root can be expressed as a (rational) linear combinations of ...
- 1,461
6
votes
2
answers
1k
views
Conjecture on irrational algebraic numbers
Conjecture:
For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$.
Questions:
Has this conjecture been ...
- 595
6
votes
1
answer
516
views
Is every polynomial a factor of a trinomial?
We say that $T(X) \in \mathbb{Q}[X]$ is a trinomial if there exist $A,B,C \in \mathbb{Q}$ such that $T(X) = AX^n + BX^m + C$ for some $n \geq m \in \mathbb{N}$.
Is it true that for each irreducible ...
- 11.2k
5
votes
2
answers
502
views
n-th root of unity in n-th division field of abelian variety?
Let $K$ be a number field and $A/K$ an abelian variety over it.
Can it be that $K(A[n])$ does not contain a primitive $n$-th rooth of unity?
If the answer is yes is it always possible to ...
- 53
4
votes
3
answers
1k
views
Ramified primes in the Chebotarev Density Theorem
I am trying to use the Chebotarev Density Theorem to say something about the Galois groups of a class of polynomials. To be more precise, by factoring a polynomial mod some prime p, I want to show ...
- 1,327
4
votes
1
answer
407
views
Can each ideal class contain an ideal with norm equal to $1$?
Let $K$ be an imaginary quadratic number field. Let $\mathcal O$ be an order in $K$. Can it happen, that there are $h(\mathcal O)>1$ fractional proper $\mathcal O$-ideals, representing the ideal ...
- 2,335
4
votes
1
answer
495
views
How does this calculation of Siegel make sense?
I am reading Siegel's paper Zum Beweise des Starkschen Satzes. Let $K$ be an imaginary quadratic field with $d_K=-p$, $p=4k+3$ a prime, and such that $h_K=1$.
Let $f=4m+1$ be a prime inert in $K$, ...
- 2,335
4
votes
0
answers
485
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
3
votes
1
answer
307
views
How many non principal prime ideals does a number field contain?
Let $K$ be a number field with ring of integers $O_K$ is not PID. Can we estimate the cardinality of the following sets
$$\mathcal{A}= \{\mathcal{P}\subset O_K \ |\ Nm(\mathcal{P})\leq x, \mathcal{P}\...
- 723
2
votes
1
answer
266
views
On Elkies' $\text{9T32}$ nonic and a shared property with j-function formulas
I. First Set
Before going to Elkies' nonic, we start with something a bit simpler. There is a list of j-function formulas in this MSE post. For example, for prime levels $p = 5,7,13,$ we have,
$$j=\...
- 12.1k