Advanced Algebra, 7.5 Credits

Contents

The course treats several different areas in algebra, with focus on how to effectively and constructively solve different probles. In this respect, the course builds on basic concepts in abstract algebra, and a common theme is the theory of zeros of polynomials and factorization of polynomials.

In the first part of the course, we study how numbers and polynomial can be multiplied faster than by the usual method. We then treat different methods for multiplying matrices more efficiently than by the naive method. After these initial steps, we study polynomial and how to effectively factorize a polynomial and find its zeros. In the last part of the course, we investigate how to find short vectors of integers, given certain conditions on the integers, and how this ties in with the theory of polynomials.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• define basic ypes of factorizations of polynomials
• define and account for the theoretical concepts underlying Hensel lifting
• prove and give detailed explanations of theorems treated

Skills and abilities

• apply methods for factoring polynomials over finite field
• apply factorization to find the zeros of a polynomial over a field
• perform multiplication of matrices using Strassen's method
• perform fast multiplication of polynomials using Karatsuba's method
• decide if a polynomial is irreducible over a given finite field
• determine the dimension of a vector lattice
• determine if a lattice basis is reduced using the LLL-method

Judgment and approach

• evaluate the effectiveness of the methods treated

Required Knowledge

The course requires a total of 90 ECTS of which 60 ECTS is within Mathematics including a course in Algebraic Structures minimum 7,5 ECTS. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The mode of teaching is mainly lectures.

Examination modes

The course is examined through written hand-in exercises. The course is graded by Fail (U), Pass (G) or Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications
 

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics.