CC220703 Introduction to Algebra

Hiroshima University Syllabus

Japanese

Academic Year 2024Year School/Graduate School School of Education

Lecture Code CC220703 Subject Classification Specialized Education

Subject Name 代数学概論

Subject Name
（Katakana）ダイスウガクガイロン

Subject Name in
English Introduction to Algebra

Instructor KITADAI YUKINORI

Instructor
(Katakana) キタダイ　ユキノリ

Campus Higashi-Hiroshima Semester/Term 2nd-Year,  First Semester,  2Term

Days, Periods, and Classrooms (2T) Thur1-4：EDU L104

Lesson Style Lecture Lesson Style
(More Details) 　

Lecture-oriented, Note-taking

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

Course Level 2 : Undergraduate Low-Intermediate

Course Area（Area） 25 : Science and Technology

Course Area（Discipline） 01 : Mathematics/Statistics

Eligible Students Required for students of Program in Mathematics Education

Keywords Integer, congruent expression, remainder
Algebra, integers, polynomials, divisors, multiples, remainder, Euclidean algeorithm, algebraic equations, groups, rings, fields

Special Subject for Teacher Education 　 Special Subject 　

Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students) Basic specialized subject
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) Integrated Arts and Sciences
（Knowledge and Understanding）
・Knowledge and understanding of the importance and characteristics of each discipline and basic theoretical framework.
（Abilities and Skills）
・The ability and skills to specify necessary theories and methods for consideration of issues.

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 several properties of integers, polynomials, divisors, multiples, remainder, Euclidean algorithm, congruences of integers, and their practical applications
- To learn concepts of the modern algebra including group theory, ring theory, and field theory throughout above materials including integers and polynomials.

Class Schedule Lesson 1: A survey of the contents about algebra in elementary schools, junior high schools, and high schools / algebra
Lesson 2: Divisors, multiples
Lesson 3:
Prime numbers
Lesson 4: Uniqueness of prime factorization
Lesson 5: Divisions with remainders, Euclidean algorithm for integers
Lesson 6: Extended Euclidean algorithm
Lesson 7: Congruences of integers, modulo calculation, congruence
Lesson 8:
Equivalence relations, quotient sets, operations on quotient sets, well-definedness
Lesson 9: Fermat's little theorem and Euler's theorem
Lesson 10: Periods of repeating decimals and congruences
Lesson 11: Chinese remainder theorem
Lesson 12: Quadratic reciprocity, divisors, multiples, divisions with remainders, Euclidean algorithm for polynomials, introduction to ring theory
Lesson 13: Advanced topic (1) - Algebraic equations and their solutions, introduction to Galois theory and group theory, the rational number field, distance, p-adic metric, Cauchy sequences, completion, the real number field and p-adic number field
Lesson 14: Advanced topic (2) - What are natural numbers and integers by the way?
Lesson 15: Advanced topic (3) - Cryptography: Caesar cipher, affince cipher and RSA

Lesson 16: Exam

There will be report assignments and quizes.

Text/Reference
Books,etc. No textbook. Students should take notes well in class．
Reference books are:
・初等整数論講義第2版, 高木貞治, 共立出版, 1971年.
・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.
・代数学1 群論入門 (代数学シリーズ), 雪江明彦, 日本評論社, 2010年.
・群論, これはおもしろい -- トランプで学ぶ群 -- (数学のかんどころ 16), 飯高茂, 共立出版, 2013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.

PC or AV used in
Class,etc. 　

(More Details) projector (computer)

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	CC220703	Subject Classification	Specialized Education
Subject Name	代数学概論
Subject Name （Katakana）	ダイスウガクガイロン
Subject Name in English	Introduction to Algebra
Instructor	KITADAI YUKINORI
Instructor (Katakana)	キタダイ　ユキノリ
Campus	Higashi-Hiroshima	Semester/Term	2nd-Year, First Semester, 2Term
Days, Periods, and Classrooms	(2T) Thur1-4：EDU L104
Lesson Style	Lecture	Lesson Style (More Details)
Lecture-oriented, Note-taking
Credits	2.0	Class Hours/Week		Language of Instruction	J : Japanese
Course Level	2 : Undergraduate Low-Intermediate
Course Area（Area）	25 : Science and Technology
Course Area（Discipline）	01 : Mathematics/Statistics
Eligible Students	Required for students of Program in Mathematics Education
Keywords	Integer, congruent expression, remainder Algebra, integers, polynomials, divisors, multiples, remainder, Euclidean algeorithm, algebraic equations, groups, rings, fields
Special Subject for Teacher Education		Special Subject
Class Status within Educational Program (Applicable only to targeted subjects for undergraduate students)	Basic specialized subject
Criterion referenced Evaluation (Applicable only to targeted subjects for undergraduate students)	Integrated Arts and Sciences （Knowledge and Understanding）・Knowledge and understanding of the importance and characteristics of each discipline and basic theoretical framework. （Abilities and Skills）・The ability and skills to specify necessary theories and methods for consideration of issues. 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 several properties of integers, polynomials, divisors, multiples, remainder, Euclidean algorithm, congruences of integers, and their practical applications - To learn concepts of the modern algebra including group theory, ring theory, and field theory throughout above materials including integers and polynomials.
Class Schedule	Lesson 1: A survey of the contents about algebra in elementary schools, junior high schools, and high schools / algebra Lesson 2: Divisors, multiples Lesson 3: Prime numbers Lesson 4: Uniqueness of prime factorization Lesson 5: Divisions with remainders, Euclidean algorithm for integers Lesson 6: Extended Euclidean algorithm Lesson 7: Congruences of integers, modulo calculation, congruence Lesson 8: Equivalence relations, quotient sets, operations on quotient sets, well-definedness Lesson 9: Fermat's little theorem and Euler's theorem Lesson 10: Periods of repeating decimals and congruences Lesson 11: Chinese remainder theorem Lesson 12: Quadratic reciprocity, divisors, multiples, divisions with remainders, Euclidean algorithm for polynomials, introduction to ring theory Lesson 13: Advanced topic (1) - Algebraic equations and their solutions, introduction to Galois theory and group theory, the rational number field, distance, p-adic metric, Cauchy sequences, completion, the real number field and p-adic number field Lesson 14: Advanced topic (2) - What are natural numbers and integers by the way? Lesson 15: Advanced topic (3) - Cryptography: Caesar cipher, affince cipher and RSA Lesson 16: Exam There will be report assignments and quizes.
Text/Reference Books,etc.	No textbook. Students should take notes well in class． Reference books are: ・初等整数論講義第2版, 高木貞治, 共立出版, 1971年. ・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. ・代数学1 群論入門 (代数学シリーズ), 雪江明彦, 日本評論社, 2010年. ・群論, これはおもしろい -- トランプで学ぶ群 -- (数学のかんどころ 16), 飯高茂, 共立出版, 2013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.
PC or AV used in Class,etc.
(More Details)	projector (computer)
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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