| Course title | Algebra 2 |
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| Course code | KMT/AL2@ |
| Organizational form of instruction | Lecture + Exercise |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 4 |
| Language of instruction | Czech |
| Status of course | Compulsory |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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Syllabus of the course Algebra 2 1. Rings - Definition of a ring - Basic properties of rings - Commutative rings - Rings with identity - Subrings - Matrix rings 2. Integral Domains, Fields, and Skew Fields - Integral domains - Zero divisors - Fields and skew fields Characteristic of a field - Relationships between rings, integral domains, and fields 3. Polynomial Rings - Polynomials over a ring and over a field - Operations with polynomials - Degree of a polynomial - Roots of polynomials - Transcendental elements 4. Field of Fractions of an Integral Domain - Construction of the field of fractions - Construction of fractions - The field of rational numbers - Properties of the field of fractions 5. Ideals and Quotient Rings - Ideals - Generated ideals - Congruences on rings - Quotient rings - Ring homomorphisms 6. Prime and Maximal Ideals - Prime ideals - Maximal ideals - Relationship between prime ideals and quotient rings - Relationship between maximal ideals and quotient fields Basic Literature: Blažek, J. a kol.: Algebra a teoretická aritmetika I. SPN, Praha, 1983. Blažek, J. a kol.: Algebra a teoretická aritmetika II. SPN, Praha, 1985. Miroslav Haviar, Pavel Klenovčan: Basic Algebra for Future Teachers.
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| Learning activities and teaching methods |
| Lecture |
| Learning outcomes |
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Course objectives Upon successful completion of the course, the graduate will be able to: - Define and analyze rings: Understand the concept of a ring, verify its fundamental properties, and work with important examples of rings, in particular matrix rings and subrings. - Distinguish and characterize basic types of rings: Determine whether a given ring is an integral domain or a field, identify zero divisors, and work with the characteristic of a field. - Work with polynomials: Define polynomials over rings and fields, perform operations with polynomials, determine their degree, and analyze their basic properties. - Understand the concept of a transcendental element: Distinguish between algebraic and transcendental elements and understand their role in algebraic constructions. - Work with fields of fractions: Explain the construction of the field of fractions of an integral domain, work with the construction of fractions, and understand its relationship to the construction of the field of rational numbers. - Work with ideals and ring congruences: Define ideals, use congruences on rings, and construct quotient rings. - Analyze prime and maximal ideals: Understand their properties and significance in the study of ring structures and quotient rings.
Competence in the field of ring theory, polynomial theory, and ideals; upon successful completion of the course, the graduate will be able to independently solve problems involving rings, integral domains, fields, polynomials, and quotient rings, work with congruences and ideals, and analyze algebraic structures using the methods of modern algebra. The graduate will also understand the relationships among fundamental algebraic structures and their significance for other areas of mathematics. |
| Prerequisites |
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Successful completion of the course Algebra 1.
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| Assessment methods and criteria |
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Mark
During the oral examination, students are required to demonstrate an active knowledge of the definitions and theorems contained in the relevant chapters of the primary course literature: - Blažek, J. a kol.: Algebra a teoretická aritmetika I. SPN, Praha, 1983. - Blažek, J. a kol.: Algebra a teoretická aritmetika II. SPN, Praha, 1985. The examination also includes solving a randomly selected exercise chosen from those provided at the end of the individual chapters covered during the course in the primary literature. To successfully pass the examination, it is therefore highly desirable to have solved all of these exercises in advance, either through independent study or within seminars and exercise sessions. |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB25) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB25) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB23) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB19) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB26) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB26) | Category: Pedagogy, teacher training and social care | 1 | Recommended year of study:1, Recommended semester: Summer |