# Tagged Questions

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 ...

9k 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 ...
9k 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 ...
5k 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 ...
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 ...
4k views

3k views

### Applications of the Cayley-Hamilton theorem

The Cayley-Hamilton theorem is usually presented in standard undergraduate courses in linear algebra as an important result. Recall that it says that any square matrix is a "root" of its own ...
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 ...
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 ...
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 $\widetilde{\mathbf Q}=\overline{\mathbf Q}$? ...
1k 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 ...
979 views

### Simple argument regarding sums of two units in a number field?

I wonder if it is possible to show, without using the Schmidt subspace/Roth theorem/Baker's bounds on linear forms in logarithms or other very deep results, that, in a number field, not all integral ...
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 ...
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 ...
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. ...
2k 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 ...
3k 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 ...
8k views

### Practical applications of algebraic number theory?

I'm interested in learning about any applications, the more worldly the better*. Pointing to a nice reference on the number field sieve, for example, would be fine. However, let me mention one ...
1k 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 ...
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 in...)....
1k views

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 ...
801 views

### Elementary congruences and L-functions

In a recent article, Emmanuel Lecouturier proves a generalization of the following surprising result: for a Mersenne prime $N = 2^p - 1 \ge 31$, the element $$S = \prod_{k=1}^{\frac{N-1}2} k^k$$ ...
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 ...
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 $\mathbb{K}$, $a$...
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 ...
991 views

### Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...
3k 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 ...
1k views

### What's special about the circle problem?

Let $K$ be a number field, and let $$\zeta_{K}(s):= \sum_{0 \neq I \text{ ideal of }O_K} \frac{1}{N_{K/\mathbb{Q}}(I)^s} = \sum_{n \ge 1} \frac{a_n}{n^s}$$ be the Dedekind zeta function of $K$. The ...
490 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 ...
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 ...
563 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 ...
964 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 ...
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 ...
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 $K$ ...