{"title":"正特征域上有限阶高导的若干注释","authors":"H. Yanagihara","doi":"10.32917/hmj/1206138808","DOIUrl":null,"url":null,"abstract":"In this short note we give a generalization of an approximation theorem on iterated higher derivations given by F. K. Schmidt in a paper [_2J (see Satz 14). Our generalization is done by determining all the iterated higher derivations of finite rank in any field K of a positive characteristic p. The following result on a derivation d in K will play an essential role in the proof: if we have d = 0, then d~\\a) = 0 if and only if a = d(β) for some β in K*\\ We shall give a proof of this fact using the Jacobson-Bourbaki's theorem which asserts the existence of a 1 — 1 correspondence between subfields of finite codimension in a field K and certain subrings of the ring M(K) of endomorphisms of the additive group (K, +) . Lastly we shall be concerned with conditions for a purely inseparable extension K of finite degree over a field k to be a tensor product of simple extensions over k. These conditions will be given in terms of higher derivations in K.","PeriodicalId":17080,"journal":{"name":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","volume":"25 1","pages":"167-171"},"PeriodicalIF":0.0000,"publicationDate":"1968-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some remarks on higher derivations of finite rank in a field of a positive characteristic\",\"authors\":\"H. Yanagihara\",\"doi\":\"10.32917/hmj/1206138808\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this short note we give a generalization of an approximation theorem on iterated higher derivations given by F. K. Schmidt in a paper [_2J (see Satz 14). Our generalization is done by determining all the iterated higher derivations of finite rank in any field K of a positive characteristic p. The following result on a derivation d in K will play an essential role in the proof: if we have d = 0, then d~\\\\a) = 0 if and only if a = d(β) for some β in K*\\\\ We shall give a proof of this fact using the Jacobson-Bourbaki's theorem which asserts the existence of a 1 — 1 correspondence between subfields of finite codimension in a field K and certain subrings of the ring M(K) of endomorphisms of the additive group (K, +) . Lastly we shall be concerned with conditions for a purely inseparable extension K of finite degree over a field k to be a tensor product of simple extensions over k. These conditions will be given in terms of higher derivations in K.\",\"PeriodicalId\":17080,\"journal\":{\"name\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"volume\":\"25 1\",\"pages\":\"167-171\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1968-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.32917/hmj/1206138808\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.32917/hmj/1206138808","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Some remarks on higher derivations of finite rank in a field of a positive characteristic
In this short note we give a generalization of an approximation theorem on iterated higher derivations given by F. K. Schmidt in a paper [_2J (see Satz 14). Our generalization is done by determining all the iterated higher derivations of finite rank in any field K of a positive characteristic p. The following result on a derivation d in K will play an essential role in the proof: if we have d = 0, then d~\a) = 0 if and only if a = d(β) for some β in K*\ We shall give a proof of this fact using the Jacobson-Bourbaki's theorem which asserts the existence of a 1 — 1 correspondence between subfields of finite codimension in a field K and certain subrings of the ring M(K) of endomorphisms of the additive group (K, +) . Lastly we shall be concerned with conditions for a purely inseparable extension K of finite degree over a field k to be a tensor product of simple extensions over k. These conditions will be given in terms of higher derivations in K.