Page 81 - Handbook Bachelor Degree of Science Academic Session 20212022

Faculty of Science Handbook, Academic Session 2021/2022

References:
SIM3003 NUMBER THEORY
1. Zhang, F.Z. (2011). Matrix theory: basic results and
nd
techniques (2 ed.). New York: Springer-Verlag.
Prime Numbers. The Division Algorithm and Unique nd
Factorization Theorem for Integers. Linear Diophantine 2. Horn, H., John, C.R. (2013). Matrix analysis (2 ed.).
Equations. Theory of congruence and the Chinese 3. Cambridge, NY: Cambridge University Press.
Zhan, X.Z. (2013). Matrix theory. Providence, RI:
Remainder Theorem. RSA encryption. Quadratic reciprocity American Mathematical Society.
and the Legendre symbol. Arithmetic functions. Primitive 4. Gentle, J.E. (2017). Matrix algebra: theory,
roots. nd
computations and applications in statistics (2 ed.).
New York: Springer-Verlag.
Assessment: 5. Randall E. C. (1979). Elements of the theory of
Continuous Assessment: 40% generalized inverses of matrices. Basel: Birkhäuser.
Final Examination: 60%
References:
1. D. M. Burton, Elementary Number Theory, 7th ed., SIM3006 ALGEBRA II
McGraw-Hill, 2011. This is a second course in abstract algebra and will cover
2. J. Silverman, Friendly Introduction to Number Theory,
A (Classic Version), 4th ed., Pearson Addison Wesley, more advanced topics on groups and rings. Topics on
2018. groups include the isomorphism theorems, various
3. R. M. Hill, Introduction to Number Theory, Essential subgroups such as the centre and commutator subgroups,
finitely generated abelian groups, automorphism groups,
Textbooks in Mathematics, World Scientific permutation groups, and p-groups.
Publishing, 2018.
4. B. Hutz, An Experimental Introduction to Number
Theory, Pure and Applied Undergraduate Texts, For rings, the focus is on commutative rings. Topics on rings
American Mathematical Society, 2018. include the maximal and prime ideals, polynomial rings,
irreducible polynomials and the Unique Factorization
Theorem.
SIM3004 ADVANCED LINEAR ALGEBRA
Assessment:
Continuous Assessment: 40%
Inner product spaces, the Cauchy-Schwarz inequality, the Final Examination: 60%
Gram-Schmidt orthogonalization process, orthogonal
complements, orthogonal projections. Adjoint operators,
normal operators, self-adjoint operators, unitary operators, References: th
positive definite operators. Bilinear forms, congruence, rank, 1. Fraleigh, J.B. First Course in Abstract Algebra, 8
Sylvester’s law of inertia, classification of symmetric bilinear 2. edition, Pearson eText, 2019. Abstract Algebra,
J.
Gallian,
Contemporary
forms, real quadratic forms. The Schur triangularization Brooks/Cole Cengage Learning, 8 edition, 2013.
th
theorem, the primary decomposition theorem, the Jordan
canonical form. 3. Hungerford, T. W. Abstract Algebra: An Introduction,
3rd edition, Brooks/Cole Cengage Learning, 2014.
4. Judson, T.W. Abstract Algebra, Theory and
Assessment: Applications, Open Source, 2019.
Continuous Assessment: 40%
Final Examination: 60%
SIM3007 RING THEORY
References:
1. Friedberg, S.H., Insel, A. J., Spence, L.E. (2019). This course includes both commutative and non-
th
Linear Algebra (5 ed.). New Jersey: Pearson commutative rings. Topics that will be discussed include
Education. subrings, subfields and ideals; internal direct sum and
2. Hoffman, K. M., Kunze, R. (1971). Linear Algebra (2 nd
ed.). New Jersey: Prentice Hall. external direct product; nil ideals, nilpotent ideals; modules
3. Cooperstein, B.N. (2015). Advanced Linear Algebra and submodules; prime ideals, maximal ideals; prime radical
and Jacobson radical; semiprime and semiprimitive rings;
nd
(2 ed.). Boca Raton: CRC Press. rings with chain conditions; group rings.
rd
4. Roman, S. (2008). Advanced Linear Algebra (3 ed.).
New York: Springer-Verlag.
5. Weintraub, S. H. (2011). A Guide to Advanced Linear Assessment:
Algebra. Washington, DC: The Mathematical Continuous Assessment: 40%
Final Examination:
60%
Association of America.
References:
1. P.M. Cohn, Introduction to Ring Theory, Springer
SIM3005 MATRIX THEORY
Undergraduate Mathematics Series, 2002.
2. I.N. Herstein, Noncommutative Rings, Carus
Rank and nullity of matrices, Sylvester’s law inequality, the Mathematical Monographs No. 15, Math Assoc. of
Frobenius inner product, the Gram-Schmidt process, the
continuity argument. Rank and full rank decompositions, LU America, 2005.
and QR decompositions, spectral decompositions, singular 3. J.A. Beachy, Introductory Lectures on Rings and
Modules, London Maths. Soc. Student Texts 47,
value decompositions, polar decompositions, Cholesky Cambridge University Press, 2012.
decompositions. Generalized inverses, Moore-Penrose 4. Fraleigh, J.B. First Course in Abstract Algebra, 8 th
inverses, the best approximation solutions, least squares
solutions. Kronecker products of matrices, permutations, edition, Pearson eText, 2019.
matrix functions of Kronecker products, Schmidt rank and 5. T.Y. Lam, Exercises in Classical Ring Theory
decompositions. (Problem Books in Mathematics), Springer, Second
Edition, 2010.
Assessment:
Continuous Assessment: 40%
Final Examination: 60%
80