# Algebra

## Objectives and Content

Objectives:

The course aims to introduce the student to the world of algebraic structures, explaining also their origin and motivation.

Content:

The primary emphasis will be on group theory (finite groups, permutation groups, subgroups, homomorphism, normal subgroup quotient, classification of finite abelian groups, symmetries group and group action) and commutative ring theory (ring, homomorphism, ideals, quotient, polynomial ring, field extension, and geometric constructions)

## Learning Outcomes

On completion of the course the student should have the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge

The student

• knows the key definitions of algebraic objects such as groups, rings and fields
• knows the main property of the aforementioned algebraic object
• know the statements (and sketch of the relative proofs) of the main theorems concerning these objects

Skills

The student

• will be able to, starting from the axiomatic definitions encountered in the course, recognize if a given algebraic structure is a group, a ring or a field, and if a subset of the given structure is a subgroup, a subring, an ideal or a subfield.
• will be able to, starting from the axiomatic definitions encountered in the course, to tell if a given map between algebraic structures is a group homomorphism or a ring homomorphism.
• Will be able to compute explicitly quotient structures.
• will be able to make explicit computation on specific examples. For example they will be able to compute the order of a element in a group, the number of orbits of a given group action, find the inverse of an element of a field, discern the feasibility of a given geometric construction.
• Will be able to apply the knowledge gained in this class to make short rigorous proofs of statement concerning the algebraic structure studied.
• Will be able to discern if two groups are or not isomorphic, with specific emphasis on finite abelian groups.

General competence

The student

• will learn to think as a mathematician. More precisely will develop the thought process necessary for any further study in the field of pure math
• will develop the specific competencies a necessary for further study or work in the academic areas requiring abstract algebra as a prerequisite

## Semester of Instruction

Spring
Required Previous Knowledge
None
Recommended Previous Knowledge
MAT121 Linear Algebra
Credit Reduction due to Course Overlap
MAUMAT644: 10 ECTS
Compulsory Assignments and Attendance
Excercises
Forms of Assessment
Written examination: 4 hours. Examination support materials: Non- programmable calculator, according to model listed in faculty regulations