Lecture # 8 and 9 : Ideals and the Class Number Formula
نویسنده
چکیده
There’s a different way to go about the problem of finding all ways of writing n = x + y. Recall that in the Gaussian integers Z[i] we have an automorphism α 7→ ᾱ. Thus the function Nα = αᾱ = x + y is multiplicative. Thus our question is just to find all ways of writing n as a norm from the Gaussian integers. To answer this question we need to know a little about the structure of Z[i]. (Note that by inducting on the norm every number factors into a product of primes.)
منابع مشابه
Weakly irreducible ideals
Let $R$ be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. The relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...
متن کامل18.785F17 Number Theory I Lecture 19 Notes: The Analytic Class Number Formula
where a ranges over nonzero ideals of OK and p ranges over nonzero prime ideals of OK ; as we showed in the previous lecture the sum and product converge absolutely on Re(s) > 1. The following theorem is often attributed to Dirichlet, although he originally proved it only for quadratic fields (this is all he needed to prove his theorem on primes in arithmetic progressions, but we will use it in...
متن کاملMathematics 129, Spring 2009 Number Fields
1. Concerning These Notes 2 2. Introduction (First Day) 2 3. Algebraic Numbers and Integers 5 4. Linear Algebra for Number Theory 10 5. Review of Field Theory; Traces/Norms 10 6. Units: Some Notes Missing Here 16 7. Diophantine Approximation 16 8. The Trace Pairing 18 9. Linear Algebra and Discriminants 21 10. The Ring of Integers Inside the Number Field 26 11. Some Computational Aspects of Dis...
متن کاملCombinatorics of Triangulations and Hilbert Series
We introduce the basic concepts of Gröbner basis theory and its relations to polytope theory. This will cover very basic parts of [1], the basic chapters from [3] and the first chapters from [11]. In particular, we will define the Gröbner fan, which will play a major role in some of the research problems we will pose later. In this lecture we relate triangulations to Gröbner bases and Hilbert-s...
متن کاملON SPECIAL p-BOREL FIXED IDEALS
We define the reduced horseshoe resolution and the notion of conjoined pairs of ideals in order to study the minimal graded free resolution of a class of p-Borel ideals and recover Pardue’s regularity formula for them. It will follow from our technique that the graded betti numbers of these ideals do not depend on the characteristic of the base field k.
متن کامل