Syllabus for Algebra II

Algebra II

Syllabus

  • 5 credits
  • Course code: 1MA006
  • Education cycle: First cycle
  • Main field(s) of study and in-depth level: Mathematics G1F

    Explanation of codes

    The code indicates the education cycle and in-depth level of the course in relation to other courses within the same main field of study according to the requirements for general degrees:

    First cycle
    G1N: has only upper-secondary level entry requirements
    G1F: has less than 60 credits in first-cycle course/s as entry requirements
    G1E: contains specially designed degree project for Higher Education Diploma
    G2F: has at least 60 credits in first-cycle course/s as entry requirements
    G2E: has at least 60 credits in first-cycle course/s as entry requirements, contains degree project for Bachelor of Arts/Bachelor of Science
    GXX: in-depth level of the course cannot be classified.

    Second cycle
    A1N: has only first-cycle course/s as entry requirements
    A1F: has second-cycle course/s as entry requirements
    A1E: contains degree project for Master of Arts/Master of Science (60 credits)
    A2E: contains degree project for Master of Arts/Master of Science (120 credits)
    AXX: in-depth level of the course cannot be classified.

  • Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
  • Established: 2007-03-15
  • Established by:
  • Revised: 2020-02-10
  • Revised by: The Faculty Board of Science and Technology
  • Applies from: week 27, 2020
  • Entry requirements: Algebra I.
  • Responsible department: Department of Mathematics

Learning outcomes

On completion of the course, the student should be able to:

  • give an account of important concepts and definitions in the theory of rings and fields;
  • exemplify and interpret important concepts in specific cases;
  • formulate important results and theorems covered by the course;
  • describe the main features of the proofs of important theorems;
  • use the theory, methods and techniques of the course to solve simple number theoretic problems and problems about rings and fields;
  • present mathematical arguments to others.

Content

Number theory: Congruences, Euler's phi-function, Fermat's little theorem, linear congruences, , the RSA algorithm.
An introduction to ring and field theory: Properties of addition and multiplication in Z, Q, R, Z[x] and C[x]. The ring and field concepts. Invertible elements and prime elements. Unique factorisation in Z and in K[x]. The Euclidean ring notion, unique factorisation and the ring Z[i] of Gaussian integers. Isomorphism, homomorphism, ideal, quotient field. The ring Z_n of integers modulo n. Chinese remainder Theorem as a ring isomorphism. Examples of non-commutative rings.

Instruction

Lectures and problem solving sessions.

Assessment

Written (4 Credit Points) and oral (1 Credit points) examination.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.

Reading list

Reading list

Applies from: week 26, 2020

Some titles may be available electronically through the University library.

  • Björklund, Johan; Hedén, Isac Algebra II : Kompendium

    Uppsala: Matematiska institutionen, Uppsala universitet,

    Find in the library

    Mandatory