All the Math You Missed最新文献

Differential Forms and Stokes’ Theorem 微分形式和斯托克斯定理

All the Math You Missed Pub Date : 2021-07-01 DOI: 10.1017/9781108992879.007

T. Garrity

引用次数: 0

Combinatorics and Probability Theory 组合与概率论

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.019

引用次数: 0

Index 指数

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.024

引用次数: 0

Elementary Number Theory 初等数论

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.011

Thomas Garrity

{"title":"Elementary Number Theory","authors":"Thomas Garrity","doi":"10.1017/9781108992879.011","DOIUrl":"https://doi.org/10.1017/9781108992879.011","url":null,"abstract":"9 + i = (5 + 3i)(1− i) + (1 + 3i) 5 + 3i = (1 + 3i)(1− i) + 1 + i 1 + 3i = (1 + i)(2 + i) Thus gcd(9 + i, 5 + 3i) = 1 + i. To find the Bezout representation, we compute 1 + i = 5 + 3i− (1 + 3i)(1− i) = 5 + 3i− (9 + i− (5 + 3i)(1− i))(1− i) = (5 + 3i)(1− 2i)− (9 + i)(1− i). Thus 1 + i = (5 + 3i)(1− 2i)− (9 + i)(1− i). (2) (a) Show that {0, 1, i, 1+ i} is a complete system of residue classes modulo 2 in Z[i]. Let α = a+bi. Reduction modulo 2 shows that α ≡ 0, 1, i, 1+ i mod 2. Moreover, none of these residue classes are equal since their differences 1, i, 1 + i are not divisible by 2. (b) Show that α ≡ 0, 1 mod 2 for every α ∈ Z[i]. The squares of the residue classes 0, 1, i, 1 + i mod 2 are 0 or 1 mod 2. Or directly: (a + bi) = a − b + 2abi ≡ a − b ≡ 0, 1 mod 2. (c) Assume that π ∈ Z[i] has odd norm and can be written in the form π = α + β for α, β ∈ Z[i]. Show that π ≡ 1 mod 2. We have π = α + β ≡ 0, 1, 2 mod 4; since Nπ is odd, we must have π ≡ 1 mod 2. (d) Show that any π ∈ Z[i] with π ≡ 1 mod 2 can be written in the form π = α + β for α, β ∈ Z[i]. We have π = α + β = (α + iβ)(α − iβ). From α + iβ = π and α− iβ = 1 we get α = π+1 2 and β = π−1 2i , and since π ≡ 1 mod 2 these are Gaussian integers. (e) Write 1 + 4i as a sum of two squares in Z[i]. 1 + 4i = (1 + 2i) + 2.","PeriodicalId":137667,"journal":{"name":"All the Math You Missed","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130505346","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

Countability and the Axiom of Choice 可数性与选择公理

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.010

引用次数: 0

Calculus for Vector-Valued Functions 向量值函数的微积分

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.004

T. Garrity

引用次数: 0

Classical Stokes’ Theorems 经典的斯托克斯定理

All the Math You Missed Pub Date : 2021-06-30 DOI: 10.1017/9781108992879.006

引用次数: 0

Curvature for Curves and Surfaces 曲线和曲面的曲率