{"title":"Carlitz素数扭拓的整数导数基","authors":"A. Maurischat, R. Perkins","doi":"10.5802/pmb.32","DOIUrl":null,"url":null,"abstract":"Let $\\mathfrak{p}$ be a monic irreducible polynomial in $A:=\\mathbb{F}_q[\\theta]$, the ring of polynomials in the indeterminate $\\theta$ over the finite field $\\mathbb{F}_q$, and let $\\zeta$ be a root of $\\mathfrak{p}$ in an algebraic closure of $\\mathbb{F}_q(\\theta)$. For each positive integer $n$, let $\\lambda_n$ be a generator of the $A$-module of Carlitz $\\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\\zeta,\\lambda_n] \\subset K(\\zeta, \\lambda_n)$ over $A[\\zeta] \\subset K(\\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\\omega$ evaluated at the roots of $\\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angles-Pellarin.","PeriodicalId":194637,"journal":{"name":"Publications Mathématiques de Besançon","volume":"206 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2016-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"8","resultStr":"{\"title\":\"An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions\",\"authors\":\"A. Maurischat, R. Perkins\",\"doi\":\"10.5802/pmb.32\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $\\\\mathfrak{p}$ be a monic irreducible polynomial in $A:=\\\\mathbb{F}_q[\\\\theta]$, the ring of polynomials in the indeterminate $\\\\theta$ over the finite field $\\\\mathbb{F}_q$, and let $\\\\zeta$ be a root of $\\\\mathfrak{p}$ in an algebraic closure of $\\\\mathbb{F}_q(\\\\theta)$. For each positive integer $n$, let $\\\\lambda_n$ be a generator of the $A$-module of Carlitz $\\\\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\\\\zeta,\\\\lambda_n] \\\\subset K(\\\\zeta, \\\\lambda_n)$ over $A[\\\\zeta] \\\\subset K(\\\\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\\\\omega$ evaluated at the roots of $\\\\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angles-Pellarin.\",\"PeriodicalId\":194637,\"journal\":{\"name\":\"Publications Mathématiques de Besançon\",\"volume\":\"206 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-11-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"8\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Publications Mathématiques de Besançon\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5802/pmb.32\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Publications Mathématiques de Besançon","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/pmb.32","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions
Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[\theta]$, the ring of polynomials in the indeterminate $\theta$ over the finite field $\mathbb{F}_q$, and let $\zeta$ be a root of $\mathfrak{p}$ in an algebraic closure of $\mathbb{F}_q(\theta)$. For each positive integer $n$, let $\lambda_n$ be a generator of the $A$-module of Carlitz $\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\zeta,\lambda_n] \subset K(\zeta, \lambda_n)$ over $A[\zeta] \subset K(\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\omega$ evaluated at the roots of $\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angles-Pellarin.