This book takes the reader on a journey from familiar high school
mathematics to undergraduate algebra and number theory. The journey
starts with the basic idea that new number systems arise from solving
different equations, leading to (abstract) algebra. Along this journey,
the reader will be exposed to important ideas of mathematics, and will
learn a little about how mathematics is really done.
Starting at an elementary level, the book gradually eases the reader
into the complexities of higher mathematics; in particular, the formal
structure of mathematical writing (definitions, theorems and proofs) is
introduced in simple terms. The book covers a range of topics, from the
very foundations (numbers, set theory) to basic abstract algebra
(groups, rings, fields), driven throughout by the need to understand
concrete equations and problems, such as determining which numbers are
sums of squares. Some topics usually reserved for a more advanced
audience, such as Eisenstein integers or quadratic reciprocity, are
lucidly presented in an accessible way. The book also introduces the
reader to open source software for computations, to enhance
understanding of the material and nurture basic programming skills. For
the more adventurous, a number of Outlooks included in the text offer a
glimpse of possible mathematical excursions.
This book supports readers in transition from high school to university
mathematics, and will also benefit university students keen to explore
the beginnings of algebraic number theory. It can be read either on its
own or as a supporting text for first courses in algebra or number
theory, and can also be used for a topics course on Diophantine
equations.
