**64**

votes

**10**answers

8k 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 ...

**64**

votes

**4**answers

8k views

### 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 ...

**44**

votes

**5**answers

2k 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 ...

**42**

votes

**13**answers

4k views

### Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...

**37**

votes

**7**answers

3k views

### How to picture $\mathbb{C}_p$?

I hope this is appropriate for mathoverflow. Understanding $\mathbb{C}_p$ has always been something of a stumbling block for me. A standard thing to do in number theory is to take the completion ...

**36**

votes

**3**answers

2k views

### 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 ...

**36**

votes

**4**answers

3k 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 ...

**35**

votes

**1**answer

1k views

### Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...

**34**

votes

**2**answers

2k 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 ...

**33**

votes

**2**answers

923 views

### What is an infinite prime in algebraic topology?

The links between algebraic topology (stable homotopy theory in particular) and number theory are nowadays abundant and fruitful. In one direction, there is chromatic homotopy theory, exploiting the ...

**32**

votes

**3**answers

2k views

### If Spec Z is like a Riemann surface, what's the analogue of integration along a contour?

Rings of functions on a nonsingular algebraic curve (which, over $\mathbb{C}$, are holomorphic functions on a compact Riemann surface) and rings of integers in number fields are both examples of ...

**31**

votes

**1**answer

1k views

### Is my field algebraically closed?

For a field $L$, let $\widetilde L$ be the splitting field of all irreducible polynomials over $L$ having prime-power degree.
Question: Do we have $\widetilde{\mathbf Q}=\overline{\mathbf Q}$?
...

**31**

votes

**1**answer

3k 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 ...

**29**

votes

**3**answers

2k views

### On what kind of objects do the Galois groups act?

I am neither number theorist nor algebraic geometer. I am wondering
whether Galois groups of number fields (say the absolute Galois
group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$) act on objects which
...

**28**

votes

**1**answer

1k views

### How would Hilbert and Weber think about the Langlands programme?

Explanations to a general mathematical audience about the Langlands programme often advertise it as "non-abelian class field theory". They usually begin as follows: a modern style formulation of ...

**28**

votes

**2**answers

2k views

### Class Numbers and 163

This is a bit fluffier of a question than I usually aim for, so apologies in advance if this doesn't pass the smell test for suitability.
Likely my favorite fun fact in all of number theory is the ...

**27**

votes

**6**answers

1k 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.
...

**27**

votes

**1**answer

998 views

### What happened to Emmy Noether's *Zukunftsphantasie* ?

Recenly I came across Peter Roquette's article On the history of Artin's $L$-functions and conductors (23 July 2003) in which he talks about some letters from Emil Artin and Emmy Noether to Helmut ...

**26**

votes

**3**answers

2k views

### Why aren't there more classifying spaces in number theory?

Much of modern algebraic number theory can be phrased in the framework of group cohomology. (Okay, this is a bit of a stretch -- much of the part of algebraic number theory that I'm interested ...

**26**

votes

**1**answer

1k 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 - ...

**26**

votes

**1**answer

1k views

### Do the algebraic integers form a free abelian group?

It is a well-known fact, proved in every introductory textbook on algebraic number theory, that if $K$ is an algebraic number field, i.e. a finite extension of $\mathbb{Q}$, then its ring ...

**25**

votes

**3**answers

2k views

### Galois Groups of a family of polynomials

I've stumbled across the family of polynomials
$ f_p(x) = x^{p-1} + 2 x^{p-2} + \cdots + (p-1) x + p $,
where $p$ is an odd prime. It's not too hard to show that $f_p(x)$ is irreducible over ...

**24**

votes

**3**answers

2k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**24**

votes

**1**answer

2k views

### Ramanujan and algebraic number theory

One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
which has the ...

**24**

votes

**0**answers

842 views

### Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$,
and define the Fekete polynomials
$$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$
Define
$$ f_N(X) = ...

**23**

votes

**2**answers

2k views

### How often are irrational numbers well-approximated by rationals?

Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if ...

**22**

votes

**9**answers

2k views

### What are some interesting problems in the intersection of Algebraic Number Theory and Algebraic Topology?

I'm a beginning graduate student and while my background is primarily in algebraic number theory, I've found myself a bit smitten with the subject of algebraic topology recently after only having read ...

**22**

votes

**0**answers

673 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

**22**

votes

**0**answers

1k 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 ...

**21**

votes

**1**answer

426 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 ...

**20**

votes

**2**answers

1k views

### Are there any Hecke operators acting on an elliptic curve with additive reduction that I don't know about?

I could have made this question very brief but instead I've maximally gone the other way and explained a huge amount of background. I don't know whether I put off readers or attract them this way. The ...

**19**

votes

**4**answers

2k views

### The Riemann Hypothesis and the Langlands program

On page 263 of this book review appears the following:
Given the centrality of L-functions to the Langlands program, nothing would seem more natural (than a presentation of elementary algebraic ...

**19**

votes

**1**answer

543 views

### Décomposition des nombres premiers dans des extensions non abéliennes

Gauß famously determined the cubic character of $2$ in his Disquisitiones : $2$ is a cube modulo a prime number $p\equiv1\mod3$ if and only if $p=x^2+27y^2$ for some $x,y\in\mathbf{Z}$. This implies ...

**19**

votes

**1**answer

1k views

### Weil Conjectures for Number Fields

Let $K$ be a number field with integral basis $\{\omega_1,\ldots,\omega_n\}$.
The affine variety $A_K$ defined by
$$ N_{K/\{\mathbb Q}}(X_1 \omega_1 + \ldots + X_n \omega_n) = 1 $$
is an algebraic ...

**18**

votes

**1**answer

2k views

### Fermat's Last Theorem in the cyclotomic integers.

Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am ...

**18**

votes

**3**answers

2k views

### Commutative Algebra with a View Toward Algebraic Number Theory

Someone asked me this today, and I don't know what the standard answer is:
Is there an analogue of David Eisenbud's rather amazing Commutative Algebra With a View Toward Algebraic Geometry but with a ...

**18**

votes

**1**answer

928 views

### Can Eisenstein's lattice point proof of quadratic reciprocity be generalized?

I'm referring to this proof. The key formula ("Eisenstein's Lemma") is
$$\left(\frac{q}{p}\right)=(-1)^{\sum_{u}\lfloor\frac{qu}{p}\rfloor},\text{ where $u=2,4,\ldots,p-1$}$$
The sum in the exponent ...

**18**

votes

**1**answer

1k views

### Number of distinct values taken by x^x^…^x with parentheses inserted in all possible ways

For what positive x's the number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways is not represented by the sequence A000081? Is it exactly the set of positive ...

**18**

votes

**1**answer

915 views

### What is the ring of integers of the Pythagorean field?

Following Hilbert, we call the complex numbers constructible via
compass and straight-edge the field of Euclidean numbers, and
the totally real such numbers the field of Pythagorean numbers. (Among ...

**18**

votes

**1**answer

711 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

**18**

votes

**1**answer

2k views

### Potential modularity and the Ramanujan conjecture

A little background: Let $f(z)=\sum_{n=1}^{\infty} a(n) e^{2\pi i nz}$ be a classical holomorphic cuspidal eigenform on $\Gamma_1(N)$, of weight $k \geq 2$ normalized with $a(1)=1$. The Ramanujan ...

**18**

votes

**0**answers

773 views

### Most “natural” proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...

**17**

votes

**3**answers

697 views

### What's the analogue of the Hilbert class field in the following analogy?

There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and ...

**17**

votes

**2**answers

1k views

### A question about non-norm-euclidean real quadratic fields

After lurking MO for a while, I decided I'd jump in. Why not start with a question?
Specifically, I have a question about a natural set that sort of "measures" the failure of a real quadratic field ...

**16**

votes

**4**answers

2k 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?
...

**16**

votes

**1**answer

480 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!

**16**

votes

**3**answers

2k views

### Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...

**16**

votes

**2**answers

698 views

### Are there any simple, interesting consequences to motivate the local Langlands correspondence?

Let's pretend that we know local Langlands at a fairly high level of generality... i.e. we know something along the lines of:
Let $G=\mathbf{G}(F)$ be the group of $F$-points of a connected ...

**16**

votes

**1**answer

770 views

### Cyclotomic polynomials evaluated at roots of unity

Dear MO_World,
I'm working on an ergodic theory question (about a generalization of eigenfunctions for measure-preserving transformations) and have run into a number theory question concerning ...

**16**

votes

**1**answer

439 views

### Quickest and/or most elementary proof of “principal iff splits completely”?

Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ...