Algebra today, in the author’s words \`\` plays a critical role in the development of the computer and communication technology that surround us in our daily lives.” The goal of this undergraduate book is \`\` to show that the subject is alive, vibrant, exciting, and more relevant to modern technology than it has ever been.” Accordingly the book presents the classical algebraic topics (modular arithmetic, groups, rings, fields and algebraic geometry) with a view on their applications to Computer Science, Error Correcting Codes and Cryptography and sections devoted to those applications are included through the book.
Chapter 1 studies modular arithmetic with Section 1.8 devoted to bar codes and ISBN and Section 1.12 introducing the idea of public key cryptography. Chapter 2 provides the basic ideas about rings and fields, Section 2.14 studying error correcting codes (including BCH and Reed-Solomon codes). Chapters 3 and 4 deal with group theory and Section 4.10 illustrates the topic of substitution cipher with the description of the Enigma machine. Chapter 5 studies rings of polynomials, Gröbner bases and affine varieties.
The last two chapters are devoted to elliptic curves. Chapter 6 introduces the concept of elliptic curve and its group law and states some classical results concerning elliptic curves over \(\mathbb{B}\) (theorems of Mordell-Weil, Mazur and Lutz-Nagell) and over a finite field (Hasse’s theorem). Chapter 7 gathers some further topics related to elliptic curves as elliptic curve cryptosystems, the role of elliptic curves in the Wiles’ proof of the Fermat last theorem or the Lenstra’s factoring algorithm (the author also points out the existence of elliptic primality tests but he do not detail them).
As conclusion, this book combines an introduction to abstract algebra with the presentation of some of its modern technological applications, which can contribute to awake up the interest of the students for this branch of the mathematics.