Without doubt the theory of numbers was Gauss' favourite subject. Indeed, in a much quoted dictum, he asserted that Mathematics is the Queen of the Sciences and the Theory of Numbers is the Queen of Mathematics. Moreover, in the introduction to Eisenstein's Mathematische Abhandlungen, Gauss wrote ‘The Higher Arithmetic presents us with an inexhaustible storehouse of interesting truths – of truths, too, which are not isolated but stand in the closest relation to one another, and between which, with each successive advance of the science, we continually discover new and sometimes wholly unexpected points of contact. A great part of the theories of Arithmetic derive an additional charm from the peculiarity that we easily arrive by induction at important propositions which have the stamp of simplicity upon them but the demonstration of which lies so deep as not to be discovered until after many fruitless efforts; and even then it is obtained by some tedious and artificial process while the simpler methods of proof long remain hidden from us.’

All this is well illustrated by what is perhaps Gauss' most profound publication, namely his Disquisitiones arithmeticae. It has been described, quite justifiably I believe, as the Magna Carta of Number Theory, and the depth and originality of thought manifest in this work are particularly remarkable considering that it was written when Gauss was only about eighteen years of age.

Foundations

The set 1, 2, 3, … of all natural numbers will be denoted by ℕ. There is no need to enter here into philosophical questions concerning the existence of ℕ. It will suffice to assume that it is a given set for which the Peano axioms are satisfied. They imply that addition and multiplication can be defined on ℕ such that the commutative, associative and distributive laws are valid. Further, an ordering on ℕ can be introduced so that either m < n or n < m for any distinct elements m, n in ℕ. Furthermore, it is evident from the axioms that the principle of mathematical induction holds and that every non-empty subset of ℕ has a least member. We shall frequently appeal to these properties.

As customary, we shall denote by ℤ the set of integers 0, ±1, ±2, …, and by ℚ the set of rationals, that is the numbers p/q with p in ℤ and q in ℕ. The construction, commencing with ℕ, of ℤ, ℚ and then the real and complex numbers ℝ and ℂ forms the basis of Mathematical Analysis and it is assumed known.

Division algorithm

Suppose that a, b are elements of ℕ. One says that b divides a (written b|a) if there exists an element c of ℕ such that a = bc.

Dirichlet's theorem

Diophantine approximation is concerned with the solubility of inequalities in integers. The simplest result in this field was obtained by Dirichlet in 1842. He showed that, for any real θ and any integer Q > 1 there exist integers p, q with 0 < q < Q such that |qθ − p| ≤ 1/Q.

The result can be derived at once from the so-called ‘box’ or ‘pigeon-hole’ principle. This asserts that if there are n holes containing n + 1 pigeons then there must be at least two pigeons in some hole. Consider in fact the Q + 1 numbers 0, 1, {θ}, {2θ}, …, {(Q − 1)θ}, where {x} denotes the fractional part of x as in Chapter 2. These numbers all lie in the interval [0, 1], and if one divides the latter, as clearly one can, into Q disjoint sub-intervals, each of length 1/Q, then it follows that two of the Q + 1 numbers must lie in one of the Q sub-intervals. The difference between the two numbers has the form qθ − p, where p, q are integers with 0 < q < Q, and we have |qθ − p| ≤ 1/Q, as required.

Dirichlet's theorem holds more generally for any real Q > 1; the result for non-integral Q follows from the theorem just established with Q replaced by [Q] + 1. Further it is clear that the integers p, q referred to in the theorem can be chosen to be relatively prime.

Algebraic number fields

Although we shall be concerned in the sequel only with quadratic fields, we shall nevertheless begin with a short discussion of the more general concept of an algebraic number field. The theory relating to such fields has arisen from attempts to solve Fermat's last theorem and it is one of the most beautiful and profound in mathematics.

Let α be an algebraic number with degree n and let P be the minimal polynomial for α (see § 5 of Chapter 6). By the conjugates of α we mean the zeros α1, …, αn of P. The algebraic number field k generated by α over the rationals ℚ is defined as the set of numbers Q(α), where Q(x) is any polynomial with rational coefficients; the set can be regarded as being embedded in the complex number field ℂ and thus its elements are subject to the usual operations of addition and multiplication. To verify that k is indeed a field we have to show that every non-zero element Q(α) has an inverse. Now, if P is the minimal polynomial for α as above, then P, Q are relatively prime and so there exist polynomials R, S such that PS + QR = 1 identically, that is for all x. On putting x = α this gives R(α) = 1/Q(α), as required. The field k is said to have degree n over ℚ, and one writes [k: ℚ] = n.

It has been customary in Cambridge for many years to include as part of the Mathematical Tripos a brief introductory course on the Theory of Numbers. This volume is a somewhat fuller version of the lecture notes attaching to the course as delivered by me in recent times. It has been prepared on the suggestion and with the encouragement of the University Press.

The subject has a long and distinguished history, and indeed the concepts and problems relating to the theory have been instrumental in the foundation of a large part of mathematics. The present text describes the rudiments of the field in a simple and direct manner. It is very much to be hoped that it will serve to stimulate the reader to delve into the rich literature associated with the subject and thereby to discover some of the deep and beautiful theories that have been created as a result of numerous researches over the centuries. Some guides to further study are given at the ends of the chapters. By way of introduction, there is a short account of the Disquisitiones arithmeticae of Gauss, and, to begin with, the reader can scarcely do better than to consult this famous work.

I am grateful to Mrs S. Lowe for her careful preparation of the typescript, to Mr P. Jackson for his meticulous subediting, to Dr D. J. Jackson for providing me with a computerized version of Fig. 8.1, and to Dr R. C. Mason for his help in checking the proof-sheets and for useful suggestions.