CC222405 Research Methods in Algebra I

Hiroshima University Syllabus

Japanese

Academic Year 2024Year School/Graduate School School of Education

Lecture Code CC222405 Subject Classification Specialized Education

Subject Name 代数学研究法I

Subject Name
（Katakana）ダイスウガクケンキュウホウI

Subject Name in
English Research Methods in Algebra I

Instructor KITADAI YUKINORI

Instructor
(Katakana) キタダイ　ユキノリ

Campus Higashi-Hiroshima Semester/Term 3rd-Year,  First Semester,  2Term

Days, Periods, and Classrooms (2T) Tues1-4：EDU L107

Lesson Style Lecture Lesson Style
(More Details) 　

Lecture-oriented, note-taking

Credits 2.0 Class Hours/Week 　 Language of Instruction J : Japanese

Course Level 3 : Undergraduate High-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Mainly students of the Program in Mathematics Education, Faculty of Education

Keywords algebra, integer, ring, commutative ring, non-commutative ring, ideal, module, quotient ring, homomorphism/isomorphism theorem, characteristic

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students) a major
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Secondary School Mathematics Education
（Knowledge and Understanding）
・To understand basic knowledge of teaching contents of mathematic education.
（Abilities and Skills）
・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.

Class Objectives
/Class Outline - To understand the contents about algebra in mathematics in secondary education
- To understand basic ring theory and commutative algebra
- To understand what type of rings 'sets with operations' appearing in mathematics in secondary education are
- To understand concepts of prime numbers, prime factorization, divisors and multiples via concepts of ring theory such as ideals, prime elements and UFD
- To understand Euclidean algorithm on integers by the words of ring theory
- To understand polynomials under the viewpoint of ring theory

Class Schedule Lesson 1: Review of group theory, GCD, prime factorization, the Euclidean algorithm, definition of rings
Lesson 2: Find examples of rings
Lesson 3: The ring of integers, polynomial rings, fields, integral domainsrings (2)
Lesson 4: Homomorphisms, kernels
Lesson 5: Ideals, prime ideals, maximal ideals, generators, principal ideals
Lesson 6: multiples, divisors, ideals, principal ideal domains, Euclid domains, Euclidean algorithm
Lesson 7: Review of quotient groups, quotient rings
Lesson 8: Ring homomorphism theorem, second isomorphism theorem
Lesson 9: characteristics, positive characteristics
Lesson 10: Prime ideals, maximal ideals, , the axiom of choice, Zorn's lemma, and existence of maximal ideals
Lesson 11: Quotient rings / prime ideals and prime elements of PID
Lesson 12: prime elements, irreducible elements, PID, UFD
Lesson 13: Chinese reminder theorem
Lesson 14: What are complex numbers by the way? Algebraic equations and field extensions
Lesson 15: Equations and symmetry of solutions

Lesson 16: exam

Papers will be required.

Text/Reference
Books,etc. No textbook will be used in this course.
Reference books:
・代数入門（新装版） ―群と加群 (数学シリーズ), 堀田良之, 裳華房, 2021年.
・代数系入門, 松坂和夫, 岩波書店, 1976年.
・代数学2 環と体とガロア理論 (代数学シリーズ), 雪江明彦, 日本評論社, 2010年.
・整数論1 初等整数論からp進数へ, 雪江明彦, 日本評論社, 2013年.
・代数と数論の基礎 (共立講座 21世紀の数学), 中島匠一, 共立出版, 2000年.
・環論, これはおもしろい ―素因数分解と循環小数への応用― (数学のかんどころ), 飯高茂, 共立出版, 2013年.
・復刊可換環論, 松村英之, 共立出版, 2000年.

PC or AV used in
Class,etc. 　

(More Details) Blackboard, handouts, computers

Learning techniques to be incorporated 　

Suggestions on
Preparation and
Review When reviewing, let your brain and your hands move.  Make examples.  Ask questions.

Requirements

Grading Method exam: 70%, quizzes: 15%, reports: 15%

Practical Experience

Summary of Practical Experience and Class Contents based on it

Message

Other

Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching.

Academic Year	2024Year	School/Graduate School	School of Education
Lecture Code	CC222405	Subject Classification	Specialized Education
Subject Name	代数学研究法I
Subject Name （Katakana）	ダイスウガクケンキュウホウI
Subject Name in English	Research Methods in Algebra I
Instructor	KITADAI YUKINORI
Instructor (Katakana)	キタダイ　ユキノリ
Campus	Higashi-Hiroshima	Semester/Term	3rd-Year, First Semester, 2Term
Days, Periods, and Classrooms	(2T) Tues1-4：EDU L107
Lesson Style	Lecture	Lesson Style (More Details)
Lecture-oriented, note-taking
Credits	2.0	Class Hours/Week		Language of Instruction	J : Japanese
Course Level	3 : Undergraduate High-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Mainly students of the Program in Mathematics Education, Faculty of Education
Keywords	algebra, integer, ring, commutative ring, non-commutative ring, ideal, module, quotient ring, homomorphism/isomorphism theorem, characteristic
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)	a major
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Secondary School Mathematics Education （Knowledge and Understanding）・To understand basic knowledge of teaching contents of mathematic education. （Abilities and Skills）・To acquire and utilize the ability to think mathematically about teaching contents of mathematic education such as algebra, geometry, statistics and computer.
Class Objectives /Class Outline	- To understand the contents about algebra in mathematics in secondary education - To understand basic ring theory and commutative algebra - To understand what type of rings 'sets with operations' appearing in mathematics in secondary education are - To understand concepts of prime numbers, prime factorization, divisors and multiples via concepts of ring theory such as ideals, prime elements and UFD - To understand Euclidean algorithm on integers by the words of ring theory - To understand polynomials under the viewpoint of ring theory
Class Schedule	Lesson 1: Review of group theory, GCD, prime factorization, the Euclidean algorithm, definition of rings Lesson 2: Find examples of rings Lesson 3: The ring of integers, polynomial rings, fields, integral domainsrings (2) Lesson 4: Homomorphisms, kernels Lesson 5: Ideals, prime ideals, maximal ideals, generators, principal ideals Lesson 6: multiples, divisors, ideals, principal ideal domains, Euclid domains, Euclidean algorithm Lesson 7: Review of quotient groups, quotient rings Lesson 8: Ring homomorphism theorem, second isomorphism theorem Lesson 9: characteristics, positive characteristics Lesson 10: Prime ideals, maximal ideals, , the axiom of choice, Zorn's lemma, and existence of maximal ideals Lesson 11: Quotient rings / prime ideals and prime elements of PID Lesson 12: prime elements, irreducible elements, PID, UFD Lesson 13: Chinese reminder theorem Lesson 14: What are complex numbers by the way? Algebraic equations and field extensions Lesson 15: Equations and symmetry of solutions Lesson 16: exam Papers will be required.
Text/Reference Books,etc.	No textbook will be used in this course. Reference books: ・代数入門（新装版） ―群と加群 (数学シリーズ), 堀田良之, 裳華房, 2021年. ・代数系入門, 松坂和夫, 岩波書店, 1976年. ・代数学2 環と体とガロア理論 (代数学シリーズ), 雪江明彦, 日本評論社, 2010年. ・整数論1 初等整数論からp進数へ, 雪江明彦, 日本評論社, 2013年. ・代数と数論の基礎 (共立講座 21世紀の数学), 中島匠一, 共立出版, 2000年. ・環論, これはおもしろい ―素因数分解と循環小数への応用― (数学のかんどころ), 飯高茂, 共立出版, 2013年. ・復刊可換環論, 松村英之, 共立出版, 2000年.
PC or AV used in Class,etc.
(More Details)	Blackboard, handouts, computers
Learning techniques to be incorporated
Suggestions on Preparation and Review	When reviewing, let your brain and your hands move. Make examples. Ask questions.
Requirements
Grading Method	exam: 70%, quizzes: 15%, reports: 15%
Practical Experience
Summary of Practical Experience and Class Contents based on it
Message
Other
Please fill in the class improvement questionnaire which is carried out on all classes. Instructors will reflect on your feedback and utilize the information for improving their teaching.

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