Students that take this course are expected to have a strong interest in Algebra.
The level of difficulty of this course is considerably higher than that of Numbers, Symmetries and Groups (ECM1706).
There are two pieces of assessed coursework, each counts for 10%. See the ELE page for the course
for hand in dates, exercises sheets and solutions.
Syllabus:
I. Groups.
- I.1. Basics about groups:
L1-3, p1-10: review of group axioms, subgroups and basic examples: symmetric groups, dihedral groups, cyclic groups;
L4-6, p6-10: homomorphism, kernel, image, isomorphism; the sign of a permutation, the alternating group.
- I.2. Basics about group actions:
L6-8, p11-15: group action, examples, orbit, stabiliser, the orbits form a partition, the stabilisers are subgroups, Orbit-Stabiliser Theorem;
L8-12, p15-23: Orbit Counting Lemma and examples (colourings), conjugacy classes, centre of a group, centraliser,
cycle type, the conjugacy classes of $S_n$ are labelled by partitions of $n$, direct product of groups, conjugacy classes and centralisers in $S_5$ and $A_5$.
- I.3. Further results about groups:
L13-16, p24-30: left and right cosets, normal subgroup, quotient group, the canonical homomorphism, First Isomorphism Theorem,
L16-18, p30-34: Sylow's three theorems and applications.
II. Rings and Fields.
- II.1. Basics about rings:
L19-21, p1-4: definition of a ring, commutative ring,
Examples: $\mathbb Z, \mathbb Q, \mathbb R, \mathbb C$, matrices/linear transformations, polynomials, functions, $\mathbb{Z}/n\mathbb{Z}$,
unit, field, zero divisor, integral domain, degree and leading coefficient of a polynomial, subring, field of fractions of an integral domain;
L21-23, p4-7: ideal, homomorphism, kernel, image, the characteristic of a ring, the $p$-th power map in characteristic $p$,
quotient ring, the canonical ring homomorphism, First Isomorphism Theorem for rings.
- II.2. Principal ideal domains & unique factorisation:
L23-25, p7-10: Division with remainder for polynomials, ideal generated by given ring elements, principal ideal domain (PID),
division and equivalence ( | and $\sim$), monic polynomial, gcd and lcm, Euclid's gcd algorithm for integers and polynomials ($\mathbb{Z}$ and $F[X]$ );
L26-28, p10-13: prime ideal, maximal ideal, irreducible and prime elements, unique factorisation domain (UFD),
in a PID every irreducible element is prime and every nonzero prime ideal is maximal, every PID is a UFD, gcd and lcm for any UFD;
L29-30, p13-15: Irreducibility criteria for $\mathbb Q[X]$: primitive polynomials, Gauss's Lemma and its corollaries, Eisenstein's criterion for irreducibility.
- II.3. Algebraic numbers and minimal polynomials:
L30-33, p16-18: algebraic numbers, minimal polynomial, examples (including roots of unity), the theorem on simple field extensions $F[\alpha]=F(\alpha)$,
transcendental numbers.
Prerequisites:
The literature for this course:
- J.R. Durbin, Modern Algebra: An Introduction, sixth edition, John Wiley & Sons, 2009.
- D.A.R. Wallace, Groups, Rings and Fields, Springer-Verlag London, 1998.
- P.J. Cameron, Introduction to Algebra, second edition, Oxford University Press, 2008.
- The algebra books by P.M. Cohn, S. Lang and B.L. van der Waerden.
The above books are only for supplementary reading.
I will also make hand written lecture notes available.
On successful completion of ECM2711, students will be able to:
- Give the definition of: subgroup, cyclic subgroup, normal subgroup, quotient group, homomorphism, kernel, image, isomorphism,
group action, orbit, stabiliser, the sign of a permutation, the alternating group, conjugacy class, cycle type.
- Prove elementary properties of above concepts. Check (in easy cases) whether or not a given subset of a group is a
(normal) subgroup.
- Calculate with permutations (e.g. determine the the disjoint cycle form of a permutation). State and apply the Orbit-Stabiliser Theorem,
the Orbit Counting Lemma and Sylow's Theorems.
- Give the definition of: ring, commutative ring, unit, zero divisor, subring, ideal, integral domain, field, field of fractions of an integral domain,
ring homomorphism, prime ideal, maximal ideal, the characteristic of a ring, irreducible/prime element, principal ideal domain,
unique factorisation domain, primitive polynomial, minimal polynomial.
- Prove elementary properties of above concepts. Check (in easy cases) whether or not a given set with operations is a ring and whether or not
a given subset of a ring is a subring/ideal/prime ideal/maximal ideal.
- Calculate the gcd of two polynomials using Euclid's algorithm. State and apply the First Isomorphism Theorem for rings, Gauss's lemma and
Eisenstein's criterion for irreducibility.