Introduction to Number Theory最新文献

Indefinite Forms 不确定的形式

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/04

引用次数: 0

Δ=𝑏²-4𝑎𝑐* Δ=𝑏²- 4𝑎𝑐*

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/06

引用次数: 0

Tables 表

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/07

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Quadratic Reciprocity 二次互反性

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/03

Jennifer Li

{"title":"Quadratic Reciprocity","authors":"Jennifer Li","doi":"10.1090/chel/384.h/03","DOIUrl":"https://doi.org/10.1090/chel/384.h/03","url":null,"abstract":"Quadratic Reciprocity is arguably the most important theorem taught in an elementary number theory course. Since Gauss' original 1796 proof (by induction!) appeared, more than 100 different proofs have been discovered. Here I present one proof which is not particularly well-known, due to George Rousseau [2]. (The proof was rediscovered more recently by (then) high-schooler Tim Kunisky [1].) Although not the shortest proof, it is the easiest to remember of all the elementary proofs I have encountered. In particular, it does not rely on Gauss' Lemma, or lattice counting, or Gauss sums; the only ingredients used in the proof are the Chinese Remainder Theorem, Wilson's Theorem, and Euler's Criterion. After proving Quadratic Reciprocity for the case of two odd primes, I'll show how to derive the 'supplementary' laws directly from the classical case. Let p and q be distinct odd primes. The Chinese Remainder Theorem asserts that the map σ : (Z/pqZ) × −→ (Z/pZ) × × (Z/qZ) × defined by σ(k) := (k, k) is a bijection. It follows that if we took half of (Z/pqZ) × , it would get mapped to half of (Z/pZ) × × (Z/qZ) ×. For example, set L := k ∈ (Z/pqZ) × : 1 ≤ k < pq 2 and R := (a, b) ∈ (Z/pZ) × × (Z/qZ) × : 1 ≤ b < q 2. A bit of thought shows that for each (a, b) ∈ R, there exists a unique k ∈ L such that σ(k) = ±(a, b). Thus we have (a,b)∈R (a, b) = k∈L (k, k), (1) where = ±1 and the products are taken in the group (Z/pZ) × × (Z/qZ) ×. Both sides of (1) simplify quite nicely. For brevity, set P := p−1 2 and Q := q−1 2. We have: (a,b)∈R (a, b) = a<p b<q/2","PeriodicalId":277341,"journal":{"name":"Introduction to Number Theory","volume":"93 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126212169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

The Class Group and Genera 类组和属

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/05

引用次数: 0

Prime Numbers and Unique Factorization 素数与唯一因数分解

Introduction to Number Theory Pub Date : 2018-09-26 DOI: 10.1090/chel/384.h/01

引用次数: 0