A comprehensive course in number theory (Q2895809)

The British mathematician Alan Baker (born 1939) is one of the most prominent number theorists at this time. He was awarded a Fields Medal in 1970 for his pioneering work on Diophantine equations, and he became particularly renowned through his effective methods in transcendental number theory. Among his famous treatises is the booklet `` A concise introduction to the theory of numbers'' [Cambridge etc.: Cambridge University Press. XIII, 95 p. (1984; Zbl 0554.10001)]. Published more than 25 years ago, this text of just ninety-five pages covered the material of a short preparatory course of the kind traditionally taught at Cambridge University at that time, thereby touching upon quite a variety of basic topics in number theory in a concise, yet sufficiently detailed manner.NEWLINENEWLINE NEWLINEThe book under review is a much expanded version of that primer from nearly three decades ago. In fact, the author has combined the well-tried original material with the more advanced topics from his lecture notes at a higher level. The result is the present, substantially new book which appears to be much more comprehensive, versatile, panoramic and up to date than its popular predecessor. More precisely, the author has produced an introduction to both classical and modern number theory, which now consists of seventeen chapters covering various fundamental topics.NEWLINENEWLINE NEWLINENow as before, there is the introduction to the book titled ``Gauss and Number Theory'', where a short account of the ``Disquisitiones Arithmeticae'' serves as a motivating starter. Chapter 1 treats some elementary number theory around the concepts of divisibility for integers and prime numbers. Chapter 2 discusses various basic arithmetical functions, average orders, perfect numbers, and the role of the Riemann zeta function in this context. Linear congruences are dealt with in Chapter 3, while Chapter 4 is devoted to quadratic residues and Gauss's Law of Quadratic Reciprocity. After a brief explanation of binary quadratic forms and their applications to representations of nonnegative integers as sums of squares in Chapter 5, the topic of Diophantine approximation is addresses in Chapter 6. This includes the classical relevant theorems of Dirichlet, Liouville and Minkowski, respectively, as well as a first introduction to transcendental numbers. Quadratic algebraic number fields, their units, their primes, and their factorization properties are illuminated in Chapter 7, with special emphasis on Euclidean quadratic fields and the Gaussian integers. Chapter 8 discusses several classical Diophantine equations, mainly those named after Pell, Thue, Mordell, Fermat, and Catalan, along with the famous ``\(abc\)-conjecture'' and the Mason-Stothers theorem.NEWLINENEWLINE NEWLINEThe topics of factorization of integers, primality testing, and the RSA algorithm in cryptography are briefly touched upon in Chapter 9, where also pseudoprimes, Fermat bases, and Pollard's method are taken up. Another new topic is presented in Chapter 10, which gives an introduction to general algebraic number fields and their rings of integers. Chapter 11 analyzes ideals in rings of algebraic integers, including the notion of different of a number field, whereas Chapter 12 turns to units and ideal classes, the related classical theorems, and to cyclotomic number fields. This chapter also contains a brief indication of the properties of local fields and \(p\)-adic fields as an outlook. Chapter 13 provides a first introduction to analytic number theory, that is, to the classical results on the distribution of prime numbers.NEWLINENEWLINE NEWLINEThe reader encounters here such topics as Dirichlet series, Chebyshev's approach, Mertens' results on certain sums and products involving primes, and Apéry's proof of the irrationality of the zeta value \(\zeta(3)\). Chapter 14 studies the Riemann zeta-function more closely, thereby focussing on its functional equation, its representation as a Euler product, its logarithmic derivative, and the distribution of its zeros. In this context, the Riemann hypothesis is briefly addressed, and the related Riemann-von Mangoldt formula is derived as well. The distribution of prime numbers is the theme of Chapter 15, where the celebrated prime number theorem, Dirichlet characters, Dirichlet \(L\)-functions, primes in arithmetical progressions, the class number formulae, and Siegel's theorem on Dirichlet \(L\)-functions are dealt with in detail. Chapter 16 is devoted to topics from both multiplicative and additive number theory, with sieve and circle methods playing a central role. Selberg's upper-bound sieve, its applications, the large sieve, and the Hardy-Littlewood-Vinogradov circle method in additive number theory are lucidly presented, together with an illustrating application in additive prime number theory. Chapter 17, the last section of the book, gives an introduction to elliptic curves and their significance in Diophantine geometry. The instructive account given here comprises the analytic approach via the Weierstrass \(\wp\)-function, the Mordell-Weil group, heights on elliptic curves, the Mordell-Weil theorem, conjectures on the rank of the Mordell-Weil group and their recent developments, isogenies, and endomorphisms of elliptic curves.NEWLINENEWLINE NEWLINEThe text is enhanced by numerous worked examples, further-leading exercises, and historical remarks. Also, as with the earlier version of this book from 1984, each chapter contains an extra section providing concrete hints for further reading, and there is a rich bibliography for this purpose.NEWLINENEWLINE Alan Baker's present book is another masterpiece of his expository writings in number theory. In absolutely peerless a manner, the author manages to combine a wide spectrum of topics with the highest degree of conciseness, essentiality, clarity, and user-friendliness. In this regard, the book under review is a perfect source for both students and teachers, be it for private study or for various kinds of courses in number theory.