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Results tagged with algebraic-number-theory
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user 290
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
36
votes
3
answers
3k
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 Dedek …
- 118k
33
votes
5
answers
4k
views
Is every (finite-dimensional, complex) representation of a finite group defined over the alg...
Is every (finite-dimensional, complex) representation of a finite group defined over the algebraic integers?
Apologies in advance if this is obvious.
Edit, 5/31/24: Since this question is getting some …
- 118k
21
votes
3
answers
1k
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 algebrai …
- 118k
19
votes
2
answers
2k
views
For what subsets S of (Z/nZ)* is there a Euclidean proof that there are infinitely many prim...
For small values of $n$ and $(a, n) = 1$ it is sometimes possible to give an elementary proof that there are infinitely many primes congruent to $a \bmod n$ along the lines of Euclid's classic proof o …
- 118k
18
votes
Cohen-Lenstra Heuristics reference
I don't have a book reference, but here are some rambling words about why one might, in general, expect objects $x$ to appear with probability proportional to $\frac{1}{|\text{Aut}(x)|}$. The short ve …
- 118k
17
votes
Is every (finite-dimensional, complex) representation of a finite group defined over the alg...
Edit: Ah, there's a simpler way to show that representations are defined over a number field; I've edited it in below.
19-year-old me (apparently you aren't allowed to start answers with "hello"),
I …
- 118k
13
votes
Accepted
UFD and fundamental group
It's the absolute Galois group that can be thought of as a fundamental group, since it is the étale fundamental group of $\text{Spec } K$. The ideal class group is instead a Picard group of line bundl …
- 118k
13
votes
3
answers
1k
views
Frobenius elements from the point of view of étale fundamental groups
The goal of this question is to find a "geometric" definition of Frobenius element in $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
Here are two definitions that don't work, but that should help ex …
- 118k
12
votes
Accepted
Motivation for the proof of Hilbert's Theorem 90
The map $T : a \mapsto b \sigma(a)$ is linear and has order $n$. It follows straightforwardly that $c + T c + ... + T^{n-1} c$ is a fixed point of $T$.
More generally, let $V$ be a representation of …
- 118k
10
votes
What is an infinite prime in algebraic topology?
The only result in algebraic topology I know that explicitly involves both finite and infinite primes is Dustin Clausen's lift of Hilbert reciprocity to a statement about spectra. For each prime $p$ h …
- 118k
10
votes
Analogy between the nodal cubic curve $y^2=x^3+x^2$ and the ring $\mathbb{Z}[\sqrt{-3}]$?
I wrote down the details in this blog post. Briefly, there are exactly two morphisms $\text{Spec } \mathbb{F}_4 \to \text{Spec } \mathbb{Z}[\omega]$, and their coequalizer (gluing them together) is $\ …
- 118k
9
votes
A natural way of thinking of the definition of an Artin $L$-function?
Here is a deeply ahistorical approach. Let's begin with the following topological analogue of a Galois extension with Galois group $G$, namely a Galois cover $Y \to X$ of spaces with Galois group $G$. …
- 118k
6
votes
Algebraic number theory and applications to properties of the natural numbers.
The number field sieve fits your requirements perfectly, I think.
4
votes
Accepted
Algebraic square root question
Lemma 1: If $P(x)$ is an integer polynomial with root $r$, then $P(x^2)$ is an integer polynomial with root $\sqrt{r}$.
Lemma 2: If $P(x)$ is an integer polynomial with root $r$, then $P(\sqrt{x})P …
- 118k
4
votes
Is there a notion of Galois extension for Z / p^2?
Let $R$ be a commutative ring and $G$ a finite group. A Galois extension of $R$ with Galois group $G$ is a pair consisting of a morphism $f : R \to S$ and an $R$-linear action of $G$ on $S$ such that …
- 118k