Introduction To Abstract Algebra
King's College London
Area of Study
Taught In English
Course Level Recommendations
ISA offers course level recommendations in an effort to facilitate the determination of course levels by credential evaluators.We advice each institution to have their own credentials evaluator make the final decision regrading course levels.
Recommended U.S. Semester Credits4
Recommended U.S. Quarter Units6
Hours & Credits
OverviewIntroduction To Abstract AlgebraModule code: 4CCM121ASemester: Semester 2 (spring)Module description:Aims of the course:The main aim of the course is to introduce you to basic concepts from abstract algebra, especially the notion of a group. The course will help prepare you for further study in abstract algebra as well as familiarize you with tools essential in many other areas of mathematics. The course is also intended to help you in the transitions from concrete to abstract mathematical thinking and from a purely descriptive view of mathematics to one of definition and deduction.Syllabus:? The integers: Principle of induction, Division Algorithm, greatest common divisor. Linear diophantine equations. Prime numbers, unique factorisation.? Groups: Examples - roots of unity, rotations, symmetries, dihedral groups, matrices, permutations. Group axioms and elementary properties. The order of an element and the orders of its powers. Subgroups, cosets, Lagrange's theorem. Cyclic groups, subgroups of cyclic groups. Homomorphisms, Kernels, isomorphisms, isomorphism classes of cyclic groups.? Rings: Axioms, examples and elementary properties. Group of units of a ring, units of the ring of residue classes of integers modulo n. Integral domains, fields. Homomorphism and isomorphism of rings.? Congruences: Solution of linear congruences. Simultaneous linear congruences, Chinese Remainder Theorem. Properties of the Euler function. Theorems of Euler and Fermat.Polynomials: Degree. Euclidean Algorithm, greatest common divisor. Unique factorisation theorem for polynomials over a field. Number of zeros of a polynomial over a field. Polynomials over the rationals - Gauss's lemma, Eisenstein's criterion.Suggested Reading:http://myreadinglists.kcl.ac.uk/search.html?q=ccm121Credit level: 4Credit value: 15Assessment: written examination/s; coursework;
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