{"title":"Hints for Exercises","authors":"D. Angell","doi":"10.1201/9781003111207-8","DOIUrl":"https://doi.org/10.1201/9781003111207-8","url":null,"abstract":"","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125623803","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}
{"title":"Hermite's Method for Transcendence","authors":"D. Angell","doi":"10.1201/9781003111207-5","DOIUrl":"https://doi.org/10.1201/9781003111207-5","url":null,"abstract":"","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129678980","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}
{"title":"Algebraic and Transcendental Numbers","authors":"David Angell","doi":"10.1201/9781003111207-3","DOIUrl":"https://doi.org/10.1201/9781003111207-3","url":null,"abstract":"Proof. Let A be the set of algebraic numbers. Let P be the set of polynomials with integer coefficients. We will first show that P is countable. Let Pn = polynomials of degree n or less with integer coefficients. The map Pn → Z, given by anx n + an−1x n−1 + · · ·+ a1x+ a0 7→ (an, an−1, . . . , a1, a0), is a bijection. Since Z is countable so is Pn. Now, since P = ∪n=1Pn, we know that P is countable. For every a ∈ A there exists a p ∈ P such that p(a) = 0. Let c : A → P assign to each a ∈ A a p ∈ P such that p(a) = 0. (Notice that c is a choice function.) Unfortunately, c is not one-to-one. But, we can get around this. Let R = c(A) ⊂ P , that is R is the range of c. For each p ∈ R we know that c−1(p) is a finite set. (Why?) Now A = ⋃","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121795295","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}
{"title":"Automata and Transcendence","authors":"D. Angell","doi":"10.1201/9781003111207-6","DOIUrl":"https://doi.org/10.1201/9781003111207-6","url":null,"abstract":"","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126744623","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}
{"title":"Hermite's Method","authors":"D. Angell","doi":"10.1201/9781003111207-2","DOIUrl":"https://doi.org/10.1201/9781003111207-2","url":null,"abstract":"","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131670460","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}
{"title":"Lambert's Irrationality Proofs","authors":"D. Angell","doi":"10.1201/9781003111207-7","DOIUrl":"https://doi.org/10.1201/9781003111207-7","url":null,"abstract":"","PeriodicalId":216643,"journal":{"name":"Irrationality and Transcendence in Number Theory","volume":"195 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115644091","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}
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