# Course Details for A.Y. 2014/2015

#### Name:

**Algebra Superiore 1 / Advanced Algebra 1**

### Basic information

##### Credits:

*:* Master Degree in Mathematics 6 CFU (b)

##### Degree(s):

Master Degree in Mathematics 1^{st} anno curriculum Generale Compulsory

##### Language:

English

### Course Objectives

This first module is divided in two parts: 1) Group theory, including actions of groups on sets, Sylow theorems, nilpotent groups, simple groups, alternating groups, free groups and presentations, the Todd-Coxeter algorithm, soluble groups, Schmidt-Iwasawa theorem.
2) Elemetary Galois theory, Galois correspondence andsolution of algebraic equations by radicals.

### Course Content

- Actions of groups on sets, orbits, inertia groups, orbit counting, exemples and exercises.
- Sylow theorems and applications, simple groups, nonsimplicity criteria, series, composition and principal series, the Jordan-Hoelder theorem, exemples and exercises.
- Free groups, presentations, the Todd-Coxeter algorithm, exemples and exercises.
- Nilpotent groups, central series, elementary properties of finite nilpotent groups, the Frattini argument, the Frattini subgroup and its properties, exemples and exercises.
- Soluble groups, the derived series, elementary properties, the Theorem by Schmidt Iwasawa.
- The field extension theory, Galois correspondence, normality and stability, splitting fields and splitting closures, separability, solubility of algebraic equations by radicals.

### Prerequisites and Learning Activities

An introduction to elementary abstract algebra (Sets: functions, equivalence relations and partitions, products; Groups: subgroups, homomorphism theorem and quotients, permutation groups, groups of invertible matrices; Rings, subrings ideals, quotient rings and homorphism theorem, polynomial rings, euclidean rings, PID, UFD, matrix rings; Fields: fied of fractions of a integral domain, simple extensions, finite fields.

### Assessment Methods and Criteria

Oral exam that includes the solution of problems in written form.

### Course page updates

This course page is available (with possible updates) also for the following academic years:

** To read the *** current * information on this course, if it is still available, go to the university course catalogue .
*Course information last updated on: 15 aprile 2015, 15:38*