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.
