Discourses on algebra. Transl. from the Russian by William B. Everett (Q1848023)

In this book, ``the elements of algebra as a field of contemporary mathematics are laid out, based on material bordering the school program as closely as possible.'' The author would like algebra to appeal to students in the way that Euclidean geometry does. The contents are organized around the three basic themes of numbers, polynomials and sets; while any one of these may dominate a given chapter, they are entwined to ensure a smooth transition between topics. While written for students at the secondary level, the text goes quite thoroughly into foundational matters normally encountered at university. The opening chapter treats irrationality and factorization of integers. Consideration of rational roots of polynomials over the integers leads to a chapter on divisibility of polynomials, roots, the derivative (defined algebraically) and the binomial theorem. This naturally leads to sets and combinatorics, which via the inclusion-exclusion principle gives a result about the frequency of numbers not divisible by a given set of primes and to the binomial probability distribution. We move back to number theory and the distribution of primes, and then to the real number system (defined axiomatically), limit of sequences and the location of polynomial roots using the intermediate value and Rolle's theorems. A chapter on sets discusses cardinality, the uncountability of the reals and sets of measure zero. The final chapter treats power series defined formally as well as generating functions and applications, particularly to partitions. Supplements to the chapters cover topics students may not see, even at college: Bernoulli numbers, the Chebyshev inequality for \(\pi (n)\), Sturm's theorem for locating polynomial roots, normal numbers and the Euler pentagonal theorem. There are several historical asides and a table of birth-death dates for mathematicians cited in the text. This book is highly recommended for teenagers with a strong desire to study mathematics and for secondary teachers seeking to contextualize what is on the school syllabus and to appreciate what some of their students will meet in university mathematics.