| Area of Mathematics: | Algebra and Geometry (EAG) | ||
| Semester: | 6ο | ||
| Course ID: | 62201 | ||
| Course Type: | Elective | ||
| Teaching hours per week: | Theory: 4 | Practice: 0 | Laboratory: 0 |
| ECTS : | 5 | ||
| Eclass: | For the course’s material, click here. | ||
| Instructors: |
Description
- Rings and ring characteristic, quotient field. Maximal ideals, prime ideals and quotient rings.
- Univariate polynomial rings and their ideals, division. Irreducible polynomials over Z and Q and Gauss’s lemma. Polynomial irreducibility criteria.
- Fields and field extensions, algebraic numbers. Rule and compass contructions.
- The Galois group of an extension, polynomial splitting field. Finite extensions of fields and their isomorphisms. The fundamental theorem of Galois theory.
- Finite fields and their extensions, cyclotomic poylnomials.
- Solvable groups, solvability criterion, the generic equation of degree >4 is not solvable by radicals.
- Simple extensions and characterstic.
- Applications: Solutions of equations of degree < 5 with radicals, discriminant. Generic n-degree polynomial. Regular Polygons. The fundamental theorem of Algebra
Bibliography
- J. Rotman. Galois Theory. Springer, 2012.
- Fraleigh, N. Brand. A first course in abstract algebra, 8th edition. Pearson 2020.
