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題名 有關數體的低階加羅瓦同調群
Various Results on Low Galois Gohomology Groups of Number Fields作者 康瓊如
Kan, Chiung Ju貢獻者 陳永秋
康瓊如
Kan, Chiung Ju關鍵詞 低階加羅瓦
同調
Gohomology
Galois日期 2002 上傳時間 31-Mar-2016 16:39:15 (UTC+8) 摘要 數體的低階加羅瓦同調群是最近數論裡面的一大研究主題，本文的第一部分深入討論有關這個主題的一些古典的及最近的結果以及實際的例子，一個重要的主題是有關數體的希爾伯類體所定義的擴張群是否可分解，陳氏的方法應用數結這個概念可證明一些在文獻上所得到的結果。本文深入的探討這些數結而獲得在冪零群上完成的結果。利用這方法可望推廣到可解群的問題上。
Low Galois cohomology groups of number fields are studied intensively in recent literature which in the first part of this thesis will be revisited with emphasis of giving direct and rigorous definitions and with providing concrete examples, classical and from more recent results from the literature. One important topic is the splitting problem of group extensions induced by the Hilbert class field of a given Galois extension of number fields. Following Tan`s approach which relates the splitting problem to the triviality of certain number knots, we elaborate known results on criteria for the splitting by giving new proofs which make use of number knots; we also give a complete result in the case of a nilpotent Galois extension. The case of solvable Galois extensions seems possibly settled using our approach.參考文獻 E. Artin and J. Tate, Class Field Theory, W. A. Benjamin, Inc., New York, 1967. E. Artin, Galois Theory (Notre Dame Math. Lectures, No. 2), Indiana, 1946. A. Babakhanian, Cohomological methods in group theory, Marcel Dekker, New York, 1972. M. Baker, An introduction to class field theory, Preprint, U. C. Berkeley, 1996. R. J. Bond, On the splitting of the Hilbert class field, Journal of Number Theory 42, 1992, 349-360. R. J. Bond, Some results on the capitulation problem, J. Number Theory 13, 1981, 246-254. Brumer and Rosen, Class number and ramification in number fields, Nagoya Math.J. 1963,97-101. H. Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, Berlin and New York, 1978. G. Cornell and M. Rosen, A note on the splitting of the Hilbert class field, Journal of Number Theory 28, 1988, 152-158. D. Garbanati, Extension of the Hasse norm theorem, Bull. Amer. Math. Soc. 81, 1975, 583-586. R. Gold, Hilbert class fields and split extensions, Illinois J. Math. 21, 1977, 66-69. M. Hall, The theory of groups, Macmillan, New York, 1959. F. P. Heider, Zahlentheoretische Knoten unendlicher Erweiterungen, Archiv der Math. 37, 1981, 341-352. C. S. Herz, Construction of class fields, Seminar on Complex Multiplication}, Chap. 7, Springer Verlag, New York, 1966. M. Ishida, Some unramified abelian extensions of algebraic number fields, J. Reine Angew. Math. 268/269, 1974, 165-173. K. Iwasawa, A note on the group of units of an algebraic number field, J. Math Pures Appl. 35, 1956, 189-192. J. P. Jans, Rings and Homology, Holt, Rinehart and winston, Inc. New York, 1964. W. Jehne, On knots in algebraic number theory, J. reine u. angew. Math. 311, 1979, 215-252. H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert`s theorem 94, J. Number Theory 8, 1976, 271-279. H. Kisilevsky, Some results related to Hilbert`s theorem 94, J. Number Theory 2, 1970, 199-206. F. Lemmermeyer, Construction of Hilbert 2-class fields, Preprint, Heidelberg, 1991. F. Lorenz, Ein Scholion zum Satz 90 von Hilbert, Abh. Math. Sem. Univ. Hamburg 68,1998. S. Maclane, Homology, Springer Verlag, Berlin, 1963. K. Miyake, The Capitulation Problem, Sugaku Expositions, Vol. 1, Number 2, 1988, 175-194. J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56, 1952, 294-297. E. Weiss, Cohomology of groups, Academic Press, New York, 1969. B. F. Wyman, Hilbert class fields and group extensions, Scripta Mathematica, Vol 29, 1973, 141-149. K. Yamamura, The maximal unramified extensions of the imaginary quadratic number fields with class number two, Journal of number theory 60, 1996, 42-50. 描述 碩士
國立政治大學
應用數學系
88751005資料來源 http://thesis.lib.nccu.edu.tw/record/#B2002000059 資料類型 thesis dc.contributor.advisor 陳永秋 zh_TW dc.contributor.author (Authors) 康瓊如 zh_TW dc.contributor.author (Authors) Kan, Chiung Ju en_US dc.creator (作者) 康瓊如 zh_TW dc.creator (作者) Kan, Chiung Ju en_US dc.date (日期) 2002 en_US dc.date.accessioned 31-Mar-2016 16:39:15 (UTC+8) - dc.date.available 31-Mar-2016 16:39:15 (UTC+8) - dc.date.issued (上傳時間) 31-Mar-2016 16:39:15 (UTC+8) - dc.identifier (Other Identifiers) B2002000059 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/83372 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description (描述) 88751005 zh_TW dc.description.abstract (摘要) 數體的低階加羅瓦同調群是最近數論裡面的一大研究主題，本文的第一部分深入討論有關這個主題的一些古典的及最近的結果以及實際的例子，一個重要的主題是有關數體的希爾伯類體所定義的擴張群是否可分解，陳氏的方法應用數結這個概念可證明一些在文獻上所得到的結果。本文深入的探討這些數結而獲得在冪零群上完成的結果。利用這方法可望推廣到可解群的問題上。 zh_TW dc.description.abstract (摘要) Low Galois cohomology groups of number fields are studied intensively in recent literature which in the first part of this thesis will be revisited with emphasis of giving direct and rigorous definitions and with providing concrete examples, classical and from more recent results from the literature. One important topic is the splitting problem of group extensions induced by the Hilbert class field of a given Galois extension of number fields. Following Tan`s approach which relates the splitting problem to the triviality of certain number knots, we elaborate known results on criteria for the splitting by giving new proofs which make use of number knots; we also give a complete result in the case of a nilpotent Galois extension. The case of solvable Galois extensions seems possibly settled using our approach. en_US dc.description.tableofcontents 謝詞 Abstract-----i 中文摘要-----ii Contents Introduction Part I Preliminary background on group cohomologies-----3 1 Cohomology of groups-----3 1.1 About H1(G,A)-----6 1.2 About H2(G,A)-----7 1.3 About H0(G,A) and H0(G,A)-----11 1.4 About H-1(G,A) and H-1(G,A)-----11 1.5 Snake Lemma and long exact cohomology sequences-----12 1.6 Jacques Herbrand-----17 2 Examples of low Galois cohomologies-----23 2.1 Hilbert 90 and Noether`s equation Theorem-----23 2.2 Cohomology of Units and Hilbert 94-----26 Part II Low Galois cohomologies of number fields-----31 3 Hilbert class field-----31 3.1 Prime decomposition-----31 3.2 Hilbert class field ; some examples-----34 3.3 Capitulation kernel and transfer kernel-----38 4 The problem of splitting-----44 5 Galois cohomology of Hilbert class field-----46 5.1 The proof of Wyman`s theorem 4.1-----46 5.2 Number knots of a Galois extension-----48 5.3 Wyman`s main theorem-----52 5.4 Gary Cornell`s necessary condition-----53 5.5 Robert J. Bond`s necessary and sufficient conditions in niloptent case-----56 Part III Further Directions of Research-----58 References-----60 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#B2002000059 en_US dc.subject (關鍵詞) 低階加羅瓦 zh_TW dc.subject (關鍵詞) 同調 zh_TW dc.subject (關鍵詞) Gohomology en_US dc.subject (關鍵詞) Galois en_US dc.title (題名) 有關數體的低階加羅瓦同調群 zh_TW dc.title (題名) Various Results on Low Galois Gohomology Groups of Number Fields en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) E. Artin and J. Tate, Class Field Theory, W. A. Benjamin, Inc., New York, 1967. E. Artin, Galois Theory (Notre Dame Math. Lectures, No. 2), Indiana, 1946. A. Babakhanian, Cohomological methods in group theory, Marcel Dekker, New York, 1972. M. Baker, An introduction to class field theory, Preprint, U. C. Berkeley, 1996. R. J. Bond, On the splitting of the Hilbert class field, Journal of Number Theory 42, 1992, 349-360. R. J. Bond, Some results on the capitulation problem, J. Number Theory 13, 1981, 246-254. Brumer and Rosen, Class number and ramification in number fields, Nagoya Math.J. 1963,97-101. H. Cohn, A classical invitation to algebraic numbers and class fields, Universitext, Springer-Verlag, Berlin and New York, 1978. G. Cornell and M. Rosen, A note on the splitting of the Hilbert class field, Journal of Number Theory 28, 1988, 152-158. D. Garbanati, Extension of the Hasse norm theorem, Bull. Amer. Math. Soc. 81, 1975, 583-586. R. Gold, Hilbert class fields and split extensions, Illinois J. Math. 21, 1977, 66-69. M. Hall, The theory of groups, Macmillan, New York, 1959. F. P. Heider, Zahlentheoretische Knoten unendlicher Erweiterungen, Archiv der Math. 37, 1981, 341-352. C. S. Herz, Construction of class fields, Seminar on Complex Multiplication}, Chap. 7, Springer Verlag, New York, 1966. M. Ishida, Some unramified abelian extensions of algebraic number fields, J. Reine Angew. Math. 268/269, 1974, 165-173. K. Iwasawa, A note on the group of units of an algebraic number field, J. Math Pures Appl. 35, 1956, 189-192. J. P. Jans, Rings and Homology, Holt, Rinehart and winston, Inc. New York, 1964. W. Jehne, On knots in algebraic number theory, J. reine u. angew. Math. 311, 1979, 215-252. H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert`s theorem 94, J. Number Theory 8, 1976, 271-279. H. Kisilevsky, Some results related to Hilbert`s theorem 94, J. Number Theory 2, 1970, 199-206. F. Lemmermeyer, Construction of Hilbert 2-class fields, Preprint, Heidelberg, 1991. F. Lorenz, Ein Scholion zum Satz 90 von Hilbert, Abh. Math. Sem. Univ. Hamburg 68,1998. S. Maclane, Homology, Springer Verlag, Berlin, 1963. K. Miyake, The Capitulation Problem, Sugaku Expositions, Vol. 1, Number 2, 1988, 175-194. J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56, 1952, 294-297. E. Weiss, Cohomology of groups, Academic Press, New York, 1969. B. F. Wyman, Hilbert class fields and group extensions, Scripta Mathematica, Vol 29, 1973, 141-149. K. Yamamura, The maximal unramified extensions of the imaginary quadratic number fields with class number two, Journal of number theory 60, 1996, 42-50. zh_TW
