Algebraic number theory

前置知识

35 项必需的前置知识（一直追溯到底）

你将学会

• Compute rings of integers and discriminants of quadratic and cyclotomic fields.
• Prove unique factorisation of ideals in Dedekind domains and use it to factor rational primes in quadratic fields.
• State the finiteness of the class number and Dirichlet's unit theorem, and compute small examples.

定义

• A number field is a finite extension K of ℚ; its ring of integers 𝒪_K consists of the elements of K that are roots of monic polynomials with integer coefficients.
• A Dedekind domain is a Noetherian, integrally closed domain in which every non-zero prime ideal is maximal.
• The ideal class group of K is the group of fractional ideals of 𝒪_K modulo the principal ones.

定理

• In a Dedekind domain every non-zero ideal factors uniquely as a product of prime ideals.
• The ideal class group of a number field is finite.
• Dirichlet's unit theorem: 𝒪_Kˣ is isomorphic to μ_K × ℤ^(r₁ + r₂ − 1), where μ_K is the finite group of roots of unity in K and r₁, r₂ count the real embeddings and pairs of complex embeddings.
• 𝒪_K is a unique factorisation domain if and only if its class number is 1.

常见误解

• Assuming unique factorisation of elements holds in every ring of integers; ℤ[√−5] fails it, although its ideals still factor uniquely.

可选发现

• Algebraic geometry Spec 𝒪_K is a one-dimensional scheme, and the analogy between number fields and function fields of curves guides much of arithmetic geometry.
• ← Congruences

延伸阅读

• Jürgen Neukirch, Algebraic Number Theory (1999)
• Daniel A. Marcus, Number Fields (1977)