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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
1
vote
Traces of powers of integral marices
The answer to Question 1 is yes, although I don't think I can extract a reasonable bound from the argument I have in mind. First observe that the question reduces to a question about largest (in absol …
- 118k
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
2
votes
Are all Finite Subsets of Affine n-space Algebraic sets, and related question
1) The point $(p_1, ... p_n)$ is the vanishing set of the polynomials $x_i - p_i$, and a finite union of algebraic sets is algebraic.
2) Yes. This should follow concretely from results in eliminatio …
- 118k
36
votes
3
answers
3k
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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 …
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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
6
votes
Algebraic number theory and applications to properties of the natural numbers.
The number field sieve fits your requirements perfectly, I think.
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 $G$. …
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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
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
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
19
votes
2
answers
2k
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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
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 $\ …
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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
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
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