This course is a gentle introduction to number theory and abstract algebra. The course begins with the formal development of number systems, underpinning algebraic properties of natural numbers, integers, and rational numbers to the foundational Peano axioms. This provides an opportunity for the students to revise the number systems and also to write formal proofs in algebra.
Elementary number theory: Fundamental theorem of arithmetic and its proof. The proof will be emphasised using the path it takes: Euclid’s division algorithm, Bezout’s lemma, and Euclid’s lemma. Modular arithmetic, Fermat’s little theorem and Euler’s generalisation, and the Chinese remainder theorem will also be discussed.
The arithmetical notions, properties, and processes of integers naturally lead to generalisations into the arithmetic of polynomials (over a field). Ring theory not only provides a framework for the unification of arithmetic in integers and arithmetic in polynomials but also sets up for the study of arithmetic in algebraic integers. Similarities and differences between the rings of algebraic integers will be discussed.