| Course title | Algebra 3 |
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| Course code | KMT/AL3@ |
| Organizational form of instruction | Lecture + Exercise |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Winter |
| Number of ECTS credits | 3 |
| 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 3 1. Polynomials in One Variable - Definition of a polynomial - Operations with polynomials - Polynomial division - Degree of a polynomial - Polynomials over rings and fields 2. Divisibility of Polynomials - Divisibility in polynomial rings - Greatest common divisor of polynomials - Irreducible polynomials - Unique factorization of polynomials - Eisenstein's criterion 3. Roots of Polynomials - Roots and their multiplicities - Relationship between roots and coefficients - Vieta's formulas - Factorization of polynomials using roots 4. Polynomials with Integer Coefficients - Polynomials over the ring of integers - Divisibility of polynomials with integer coefficients - Irreducibility of polynomials - Eisenstein's criterion and its applications 5. Polynomial Derivatives - Formal derivative of a polynomial - Computation of polynomial derivatives - Multiple roots of polynomials - Applications of derivatives in the study of polynomials 6. Polynomials in Several Variables - Multivariable polynomials - Operations with multivariable polynomials - Degree of a polynomial - Symmetric polynomials - Basic properties of symmetric polynomials 7. Algebraic Equations - Binomial equations - Quadratic equations over the complex numbers - Cubic equations over the complex numbers - Cardano's method - Reciprocal equations - Numerical methods for solving algebraic equations 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.
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| Learning activities and teaching methods |
Lecture, Monologic Lecture(Interpretation, Training)
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| Learning outcomes |
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Course objectives Upon successful completion of the course, the graduate will be able to: - Work with polynomials in one variable: Define polynomials, perform operations with polynomials, use polynomial division algorithms, and analyze their fundamental properties. - Analyze polynomial divisibility: Determine divisibility of polynomials, find their greatest common divisors, work with irreducible polynomials, and apply unique factorization. - Work with polynomial roots: Determine roots and their multiplicities, apply Vieta's formulas, and analyze relationships between roots and coefficients. - Work with polynomials having integer coefficients: Analyze their properties, investigate irreducibility, and apply Eisenstein's criterion. - Use polynomial derivatives: Compute polynomial derivatives and apply them to the study of multiple roots and other polynomial properties. - Work with multivariable polynomials: Perform basic operations with polynomials in several variables and analyze the properties of symmetric polynomials. - Solve algebraic equations: Apply algebraic and numerical methods to solve binomial, quadratic, cubic, and reciprocal equations. - Apply methods of polynomial algebra to mathematical problems and understand their connections with other areas of algebra and mathematical analysis.
Competence in the field of polynomial algebra and algebraic equations; upon successful completion of the course, the graduate will be able to independently solve problems involving polynomials in one or several variables, their divisibility, roots, and irreducibility, analyze algebraic equations, and apply both algebraic and numerical methods for their solution. The graduate will also understand the relationships between polynomial theory, algebraic structures, and other areas of mathematics. |
| Prerequisites |
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Prerequisites To successfully complete the course, the following prerequisites are required: - Knowledge of Secondary School Mathematics: Students should have a solid foundation in secondary school mathematics, including knowledge of polynomials, functions, equations, and basic algebraic operations. - Knowledge from Algebra 1 and Algebra 2: Students must have knowledge from Algebra 1 and Algebra 2 courses. - Basic Knowledge of Mathematical Analysis: Basic knowledge of mathematical analysis will also be useful, particularly in the areas of derivatives and fundamental functions. These prerequisites will ensure that students have the necessary background to understand and apply the advanced algebraic concepts covered in the course.
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| Assessment methods and criteria |
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Student performance, Dialog, Seminar Work
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. et al.: Algebra a teoretická aritmetika I. SPN, Praha, 1983. - Blažek, J. et al.: 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 (BB23) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB19) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB26) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB25) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB25) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB26) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 2 | Recommended year of study:2, Recommended semester: Winter |