Abstract algebra is the area of mathematics that investigates algebraic structures. By defining certain operations on sets, one can construct more sophisticated objects: groups, rings, and fields. These operations unify and distinguish objects at the same time: adding matrices is similar to adding integers, while matrix multiplication is quite different from multiplication modulo n. Because structures like groups or rings are richer than sets, we cannot compare them using only their elements; we have to relate their operations as well. For this reason group and ring homomorphisms are defined. These are functions between groups or rings that "respect" their operations. This type of function is used not only to relate these objects, but also to build new ones, quotients for example.

Although at this point it may seem like the study of these new and strange objects is little more than an exercise in a mathematical fantasy world, the basic results and ideas of abstract algebra have permeated and are at the foundation of nearly every branch of mathematics.

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## Content and learning outcomes

### Course contents

Group theory: groups, permutations, homomorphisms, group actions, Lagrange's theorem, Sylow's theorems, structure of abelian groups.

Ring theory: rings, ideals, fields and field extensions, factorization, principal ideal domains, polynomial rings, rings of integers.

### Intended learning outcomes

After completing the course a student should be able to:

• use concepts, theorems and methods to solve, and present the solution of, problems in those parts of group and ring theory described by the main contents of the course,
• read and understand mathematical text,

in order to

• be able to carry out abstract reasoning about algebraic structures
• be trained in logical thinking and in constructions of mathematical proofs
• be able to recognize and use algebraic structures in engineering and science subjects and in his or her forthcoming work.

### Course Disposition

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## Literature and preparations

### Specific prerequisites

Completed basic course SF1672 Linear Algebra or SF1624 Algebra and Geometry.

### Recommended prerequisites

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### Equipment

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### Literature

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## Examination and completion

If the course is discontinued, students may request to be examined during the following two academic years.

A, B, C, D, E, FX, F

### Examination

• TEN1 - Exam, 7,5 hp, betygsskala: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

The examiner decides, in consultation with KTHs Coordinator of students with disabilities (Funka), about any customized examination for students with documented, lasting disability. The examiner may allow another form of examination for re-examination of individual students.

### Opportunity to complete the requirements via supplementary examination

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### Opportunity to raise an approved grade via renewed examination

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### Ethical approach

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

## Further information

### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web SF1678

SCI/Mathematics

Technology

First cycle