Title of the course: Numbers and Polynomials
Instructor: Prof. Alexandre Borovik
Institution: Manchester U.
Dates: 30 July - 5 August 2018
Prerequisites: Zero. Suitable for fresh entrants to university.
Level: Younger undergraduate students
Abstract: The course deals with basic number theory (that is, integers) and polynomials, as concrete, sensible, sound, familiar to students objects, but treats them with full proofs in an unifying approach.
1. Taking possibility of division (of integers and polynomials over a field) with remainder for granted, a sequence of results about greatest common divisors, uniqueness of factorisation, etc. -- up to the Chinese Reminder Theorem and Lagrange's Interpolation Formula, will be *proved*. More precisely, every theorem will be formulated in two versions: for integers and for polynomials, with only one of them being proved in a lecture, the other one left as an exercise for students.
2. This part of the course will be rounded up by an explanation that the Chinese Reminder Theorem and Lagrange's Interpolation Formula are *one and the same thing*.
3. Then complex numbers and roots of unity will be introduced, and the Fundamental Theorem of Algebra stated, unfortunately, without proof.
4. And, finally a version of the Fermat's Theorem will be proved:
The equation X^n + Y^n = Z^n, n >1, has no solutions in polynomials X = X(t), Y = Y(t), Z = Z(t) of non-zero degree, with some historical remarks.