Lecture Description
In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields. Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization. However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.This interesting area of number theory does have some serious foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
Course Index
- History of Pythagoras' theorem
- History of Pythagoras' Theorem II
- History of Greek Geometry I
- History of Greek Geometry II
- History of Greek Number Theory
- History of Greek Number Theory II
- Infinity in Greek Mathematics
- History of Number Theory and Algebra in Asia
- History of Number Theory and Algebra in Asia II
- History of Polynomial Equations
- History of Polynomial Equations II
- History of Analytic Geometry and the Continuum
- History of Analytic Geometry and the Continuum II
- History of Projective Geometry
- History of Calculus
- History of Infinite series
- Mechanics and the Solar System
- History of Non-Euclidean Geometry
- The Number Theory Revival
- Mechanics and Curves
- Complex Numbers and Algebra
- History of Differential Geometry
- History of Topology
- Hypercomplex Numbers
- History of Complex Numbers and Curves
- History of Group Theory
- History of Galois Theory I
- History of Galois Theory II
- History of Algebraic Number Theory and Rings I
- History of Algebraic Number Theory and Rings II
- Simple groups, Lie groups, and the Search for Symmetry I
- Simple groups, Lie groups, and the Search for Symmetry II
Course Description
In this course, Prof. N.J. Wildberger from UNSW provides a great overview of the history of the development of mathematics. The course roughly follows John Stillwell's book 'Mathematics and its History' (Springer, 3rd ed)Starting with the ancient Greeks, we discuss Arab, Chinese and Hindu developments, polynomial equations and algebra, analytic and projective geometry, calculus and infinite series, number theory, mechanics and curves, complex numbers and algebra, differential geometry, topology and hyperbolic geometry. This course is meant for a broad audience, not necessarily mathematics majors. All backgrounds are welcome to take the course and enjoy learning about the origins of mathematical ideas. Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. At UNSW, this is MATH3560 and GENS2005. NJ Wildberger is also the developer of Rational Trigonometry: a new and better way of learning and using trigonometry.