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Programme & Unit Catalogues

MA20217: Algebra 2B

Credits: 6
Level: Intermediate (FHEQ level 5)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: MA20217 Mandatory extra work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA10209 and take MA10210 and take MA20216
Description: Aims:
To introduce the students to basic abstract ring theory and provide a thorough structure theory of linear operators on a finite dimensional vector spaces.

Learning Outcomes:
After taking this unit, students should be able to:
* Demonstrate understanding of the basic theory of rings.
* Factorise in various integral domains they have met throughout the course and demonstrate understanding of the general theory.
* State and prove the fundamental results on the structure theory of linear operators.
* Apply the structure theory of linear operators in examples. Determine characteristic polynomials, minimal polynomials, geometric and algebraic multiplicities as well as the Jordan normal form for a given linear operator. Calculate generalised eigenspaces.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).

Content:
Elementary axiomatic theory of rings. Integral domains, fields, characteristic. Subrings and product of rings. Homomorphisms, ideals and quotient rings. Isomorphism theorems. Fields of fractions. Polynomial rings. Maximal ideals and prime ideals. Factorisation in integral domains. Unique factorisation in principal ideal domains. Eisenstein criterion and other criteria for factorisation in polynomial rings.
Revision of eigenvalues, eigenvectors and diagonalisability. Invariant subspaces and decomposition of linear operators . Minimal polynomials and the Cayley-Hamilton theorem. The primary decomposition theorem and generalised eigenspaces. Applications including calculations of powers and exponentials of matrices. Cyclic invariant subspaces. The Jordan normal form theorem. Applications.
Programme availability:

MA20217 is Compulsory on the following programmes:

Department of Computer Science
• USCM-AFB13 : BSc (hons) Computer Science with Mathematics (Full-time) - Year 2
• USCM-AKB14 : BSc (hons) Computer Science with Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USCM-AAB14 : BSc (hons) Computer Science with Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Mathematical Sciences
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 2
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 2
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 2
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2

MA20217 is Optional on the following programmes:

Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 2
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 2
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 2
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 2
Department of Physics
• USXX-AFB03 : BSc (hons) Mathematics and Physics (Full-time) - Year 3
• USXX-AKB04 : BSc (hons) Mathematics and Physics with Placement (Full-time with Thick Sandwich Placement) - Year 4
• USXX-AAB04 : BSc (hons) Mathematics and Physics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USXX-AFM01 : MSci (hons) Mathematics and Physics (Full-time) - Year 3

NB. Programmes and units are subject to change at any time, in accordance with normal University procedures.