Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MML1007 | Topics in Algebra II (for teachers education) |
| Teaching Staff in Charge |
Assoc.Prof. PELEA Cosmin Razvan, Ph.D., cpelea math.ubbcluj.ro |
| Aims |
|
Deepening the knowledge on solving some equations. Developing some problem solving skills. Deepening the knowledge on solving linear systems. Presenting some arithmetic functions. Aproaching some classical problems by using tools provided by modern algebra. Presenting some basic results regarding the roots of polynomials with coefficients in a field. |
| Content |
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I. Preliminaries
1. Simple equations in R. Methodical overview. On the solutions of the equations ax=b and xa=b in a ring. 2. (Real polynomial) functions of degree 2. Quadratic equations in R. 3. On some systems of equations of degree 1 and/or 2. Irrational equations. 4. Exponential and logarithmic equations. Trigonometric equations. II. Linear systems of equations 1. Introductive notions in linear algebra. 2. Solving the linear systems of equations. The theorems of Kronecker-Capelli, Rouche, Cramer. Gauss method. III. Algebraic equations with complex coefficients 1. Polynomials with coefficients in a field. Introductive notions. 2. Some topics of field theory. 3. The Galois group. Abel-Ruffini Theorem. 4. Polynomials with complex coefficients. D’Allembert-Gauss Theorem. Polynomials with coefficients in R, Q and Z. 5. Algebraic equations of degree 3 with complex coefficients. 6. Algebraic equations of degree 4 with complex coefficients. 7. On some algebraic equations of degree at least 3. 8. Some topics concerning the roots of a polynomial: the resultant of two polynomials, the discriminant of a polynomial, the Newton formulas. |
| References |
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1. R. Covaci - Algebra si programare liniara, Litografia UBB, Cluj-Napoca, 1986.
2. S. Crivei - Basic Abstract Algebra, Casa Cartii de Stiinta, Cluj-Napoca 2002. 3. C. Nastasescu, C. Nita – Teoria calitativa a ecuatiilor algebrice, Editura Tehnica, Bucuresti, 1979. 4. C. Nastasescu, C. Nita, Gh. Rizescu - Matematica, Manual pentru clasa a IX-a, Editura Didactica si Pedagogica, Bucuresti, 1995. 5. C. Nastasescu, C. Nita, S. Popa - Matematica, Manual pentru clasa a X-a, Editura Didactica si Pedagogica, Bucuresti, 1995. 6. C. Nastasescu, I. Stanescu, C. Nita – Matematica, Elemente de algebra superioara, Manual pentru clasa a XI-a, Editura Didactica si Pedagogica, 1995. 7. Gh. Pic – Algebra superioara, Editura Didactica si Pedagogica, Bucuresti, 1966. 8. Gh. Pic, I. Purdea - Tratat de algebra moderna, vol.1, Editura Academiei, 1977. 9. I. Purdea, C. Pelea – Probleme de algebra, Ed. Eikon, Cluj-Napoca, 2008. 10. I. Purdea, I. Pop – Algebra, Ed. GIL, Zalau, 2003. |
| Assessment |
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Homeworks (20%). Exam. (80%) |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |