Studies of Hirata separable extendions and related ring extensions

• Project/Area Number: 23540049
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Okayama University
• Principal Investigator: Ikehata Shuichi 岡山大学, 自然科学研究科, 教授 (20116429)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 平田分離拡大 / 分離拡大 / ガロア拡大 / 平田分離多項式 / 分離多項式 / ガロア多項式 / 歪多項式環 / G-Galois拡大 / 東屋多元環 / 弱分離拡大 / Ｇ-Galois拡大

Outline of Final Research Achievements

In 1979, Yoichi Miyashita gave characterizations of a separable polynomial and a Hirata separable polynomial in skew polynomial rings. The research representative and Satoshi Yamanaka gave direct and elementary proofs of these theorems.　 Takasi Nagahara made a thorough investigation of separable polynomials and Galois polynomials of degree 2 in skew polynomial rings. The research representative and Satoshi Yamanaka tried to generalize Nagahara’s results for polynomials of degree 2 to general prime degree p case, and we had some interesting results.

Research Products

• [Journal Article] On weakly separable extensions (2015)
  • Author(s): 山中聡, 池畑秀一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1964
  • Open Access / Acknowledgement Compliant
• [Journal Article] On Frobenius polynomials in skew polynomial rings (2014)
  • Author(s): 山中聡, 池畑秀一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1915
  • Open Access / Acknowledgement Compliant
• [Journal Article] On Galois polynomials in skew polynomial rings (2014)
  • Author(s): 山中聡, 池畑秀一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1873
• [Journal Article] On Galois polynomials of degree p in skew polynomial rings of derivation type (2013)
  • Author(s): Satoshi Yamanaka and Shuichi Ikehata
  • Journal Title: Southeast Asian Bulletin of Mathematics Volume: 37, no.4
  • Peer Reviewed
• [Journal Article] An alternative proof of Miyashita’s theorem in a skew polynomial ring (2012)
  • Author(s): Satoshi Yamanaka and Shuichi Ikehata
  • Journal Title: International Journal of Algebra Volume: 6, no.21 Pages: 1011-1023
  • Peer Reviewed
• [Journal Article] 歪多項式環における宮下の定理の別証明, (2012)
  • Author(s): 山中聡, 池畑秀一
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1809 Pages: 214-223
• [Journal Article] On a theorem of A. Micali and A. Paques (2011)
  • Author(s): Shuichi Ikehata
  • Journal Title: International Journal of Algebra Volume: 5 Pages: 1155-1159
  • Peer Reviewed
• [Journal Article] On Galois extensions of rings with an inner automorphism group (2011)
  • Author(s): Shuichi Ikehata
  • Journal Title: 数理解析研究所講究録（代数と言語のアルゴリズムと計算理論） Volume: 1769 Pages: 66-69
• [Presentation] Remarks on weakly separable extensions (2016)
  • Author(s): Satoshi Yamanaka, Shuichi Ikehata
  • Organizer: 代数系、論理、言語と計算機科学
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-02-16
• [Presentation] Morita equivalence in ring extension and congruence of functors (2016)
  • Author(s): Satoshi Yamanaka, Hiroaki Komatsu, Shuichi Ikehata
  • Organizer: 代数系、論理、言語と計算機科学
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-02-16
• [Presentation] On a theorem of G. Szeto and L. Xue (2015)
  • Author(s): Shuichi Ikehata
  • Organizer: The Conference on Ring theory in honor of Professor George Szeto on his retirement
  • Place of Presentation: 岡山大学環境理工学部
  • Year and Date: 2015-11-14
  • Int'l Joint Research
• [Presentation] On weakly separable extensions (2015)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 代数系・論理・言語と計算機科学の新たな接点
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-02-16
• [Presentation] Some topics on skew polynomial rings (2014)
  • Author(s): Shuichi Ikehata
  • Organizer: Algebra Colloquium
  • Place of Presentation: Bradley University
  • Year and Date: 2014-09-03
• [Presentation] On Frobenius polynomials in skew polynomial rings (2014)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 計算機科学における論理・代数・言語
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 歪多項式環と種々の分離拡大 (2013)
  • Author(s): 池畑秀一
  • Organizer: 環論研究集会
  • Place of Presentation: 大阪樟蔭女子大学関谷キャンパス
• [Presentation] On Galois polynomials in skew polynomial rings (2013)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 代数とコンピュータサイエンス
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Hirata separable polynomials and Galois polynomials in skew polynomial rings (2013)
  • Author(s): 池畑秀一
  • Organizer: Algebra Colloquium
  • Place of Presentation: Bradley University, 米国
• [Presentation] On Galois polynomials of degree p in skew polynomial rings of derivation type II (2013)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 日本数学会春季年会
  • Place of Presentation: 京都大学
• [Presentation] Hirata separable extensions and skew polynomial rings I, II, and III, (2012)
  • Author(s): 池畑秀一
  • Organizer: Conference in Ring theory
  • Place of Presentation: Sun Yat-Sen University, 中国
  • Invited
• [Presentation] On Galois polynomials of degree p in skew polynomial rings of derivation type (2012)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 九州大学
• [Presentation] 歪多項式環におけるガロア多項式について (2012)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 日本数学会中国四国支部
  • Place of Presentation: 岡山大学環境理工学部
• [Presentation] 歪多項式環における宮下の定理の別証明 (2012)
  • Author(s): 山中聡, 池畑秀一
  • Organizer: 京都大学数理解析研究所研究集会（代数系および計算機科学基礎）
  • Place of Presentation: 京都大学数理解析研究所
• [Funded Workshop] The Conference on Ring theory in honor of Professor George Szeto on his retirement (2015)
  • Place of Presentation: 岡山大学環境理工学部
  • Year and Date: 2015-11-14