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Elements of Number Theory
Solutions of equations in integers is the central problem of number theory and is the focus of this book. The amount of material is suitable for a one-semester course. The author has tried to avoid the ad hoc proofs in favor of unifying ideas that work in many situations. There are exercises at the end of almost every section, so that each new idea or proof receives immediate reinforcement.
- GenresMathematicsNumber Theory
268 pages, Paperback
First published December 3, 2010
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About the author
John Stillwell
51 books60 followersJohn Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University
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July 25, 2020
This reviewer’s background as a mathematical physicist means that his education has concentrated more on geometry and analysis than on algebra, per se. In an unexpected turn of events, he resolved a few years ago to make good on his poor preparation and to go back to the basics. For his textbooks, he selected the presentations by the well-regarded historian of mathematics John Stillwell in the Springer series of undergraduate texts in mathematics. This turns out to have been an inspired choice; Stillwell happens to be blessed with outstanding pedagogical instincts. Here, we will review Stillwell’s Elements of Number Theory.
Part of the charm of the subject is that the natural numbers and their elementary properties are simple enough to have occupied the thoughts of the men of ancient civilizations. In the first chapter, Stillwell reproduces the Plimpton clay tablet from Babylonia in the early second millenium before Christ, which contains a list of Pythagorean triples from well over a millenium before Pythagoras himself. Surprisingly, the low-lying triples such as 3,4,5 and 5,12,13 are absent; clearly, the Babylonians must have known methods of generating much larger triples, some of which Stillwell discusses in his text. The largest entries on the tablet are in the five figures. Another very ancient formula is the two-square identity, which represents the product of two sums of squares as the sum of another two squares and figures in the theory of Gaussian integers, going back, as far as we know, to the Arabic mathematician al-Khazin in the mid-tenth century, although it is probable that Diophantus and even the Babylonians were aware of it.
Stillwell means what he says in entitling his text ‘elements’. He starts out with very elementary properties of natural numbers and integers (e.g., the definition of what a prime number is) and a little overview, to be followed up upon in later chapters, of Diophantine equations, the chord method, Pythagorean triples and Gaussian integers. In the second chapter, he introduces the Euclidean algorithm for finding the greatest common denominator of a pair of numbers, proves unique prime factorization and then launches into an interesting discussion of a graphical map of relatively prime pairs or primitive vectors. Chapter three follows with the arithmetic of congruences, Fermat’s little theorem, Euler’s theorem and the concept of a primitive root. With this, he has assembled the tools for an intelligent explanation of RSA cryptography in chapter four. The reader will enjoy being able to understand this topic, so relevant to real-world applications, and to work out a few toy examples of the encryption/decryption algorithm (with unrealistically small primes, needless to say).
The next chapter, chapter five, takes up the Pell equation, Brahmagupta’s (Indian, seventh century) composition rule for obtaining new solutions from given ones and a general theory of quadratic forms due to Conway. The Pell equation is connected with extensions of the ring of integers of the form Z[√n]. This motivates the case of Gaussian integers (i.e., Z[i]), to which considerable attention is devoted in chapter six. The concept of a Gaussian prime is very interesting and related to unique prime factorization, which continues to hold for Gaussian integers, as Stillwell covers exhaustively.
The adventure gets to be exciting when one takes on other quadratic integers, in chapter seven. As one sees there, unique prime factorization holds for Z[√-2] and Z[ζ_3] but fails for Z[√-3]! One is led by this route to the analysis of algebraic factorizations of polynomial forms such as x^2+y^2 and x^3+y^3. Stillwell gives here an explicit proof of Fermat’s last theorem in the case n=3, what is perhaps the most technically difficult result in the text but not impossible to follow, with the machinery he has built up up to this point.
The next two chapters could be seen as a detour further illustrating pleasant stopping-places in number theory. Chapter eight is given over to Hurwitz’s proof using quaternions that every natural number can be expressed as the sum of four integer squares. Chapter nine goes into the law of quadratic reciprocity, Euler’s criterion and the Chinese remainder theorem (originally from the third to fourth century in China, but dressed up here in its most modern group-theoretical guise).
The last three chapters form the heart of the book and this reviewer’s motivation for reading it in the first place. All the phenomena discussed so far are summed up in the single abstract algebraic concept of a ring. After giving the ring-theoretic axioms and relating rings to their corresponding fields, Stillwell quickly specializes to the case of quadratic fields and their integers. He then pursues the concept of an ideal in a ring, certain subsets that display behavior analogous to a set of multiples of a given number and therefore may be regarded as ‘ideal numbers’, which is how Kummer first conceived of them and how Dedekind formalized the notion. Ideals formed a central part of the modern formulation of abstract algebra at the hands of Emmy Noether during the 1910’s and 1920’s, as presented in van der Waerden’s classic monograph Moderne Algebra. Once the concept has been introduced, the properties of ideals can be derived. Stillwell discusses some cases of principal ideal domains and criteria therefor, and gives as well an example of a non-principal ideal in Z[√-3]. This reviewer remembers a glaring omission in his undergraduate course on algebra: principal ideal domains were discussed without giving a single instance of how an ideal could fail to be principal! The last topic Stillwell covers in chapters eleven and twelve are multiplication of ideals, prime ideals and prime ideal factorization, in which uniqueness can be recovered at the ideal-theoretic level despite the fact that it fails at the number-theoretic level.
In all, Stillwell manages to provide a tour through a significant amount of territory in basic number theory. His pedagogical style ensures that everything he does is comparatively easy to follow and understand. What makes this textbook precious, though, are Stillwell’s commentary along the way and closing section to each chapter, in which he recounts the history of the subject in such a way as to point out just why the concepts he introduces are important. This involves a certain amount of claims about material too advanced to go into in the text itself, but these supply perspective (for instance, on how Kummer was led to conceive of his ideal numbers during the course of a faulty attempt to prove Fermat’s last theorem). The student will come away with enough of a grounding in the simple cases Stillwell does cover in his text to be startled by the amazing properties of the natural numbers and number fields that can indeed be shown to hold in a more advanced treatment of number theory, and to have some clue of the tools and strategy of proof that will be employed.
Stillwell includes some four hundred homework exercises to solidify the reader’s grasp of the material. This reviewer finds most, indeed almost all, of them not to be all that difficult, but perhaps this is a function of his approaching this undergraduate text only after first having gained a fair degree of mathematical maturity in other fields. It may be that an undergraduate venturesome enough to attempt it would find the problems to be for the most part at about the right level to engage his interest. Certainly, they are not all routine and do test one’s understanding of the subject, although Stillwell does throw in a few purely computational problems for good measure.
In view of Stillwell’s masterly pedagogical style, this textbook must be accounted a happy discovery! It could almost—but not quite—convert the reviewer into a number theorist!
Part of the charm of the subject is that the natural numbers and their elementary properties are simple enough to have occupied the thoughts of the men of ancient civilizations. In the first chapter, Stillwell reproduces the Plimpton clay tablet from Babylonia in the early second millenium before Christ, which contains a list of Pythagorean triples from well over a millenium before Pythagoras himself. Surprisingly, the low-lying triples such as 3,4,5 and 5,12,13 are absent; clearly, the Babylonians must have known methods of generating much larger triples, some of which Stillwell discusses in his text. The largest entries on the tablet are in the five figures. Another very ancient formula is the two-square identity, which represents the product of two sums of squares as the sum of another two squares and figures in the theory of Gaussian integers, going back, as far as we know, to the Arabic mathematician al-Khazin in the mid-tenth century, although it is probable that Diophantus and even the Babylonians were aware of it.
Stillwell means what he says in entitling his text ‘elements’. He starts out with very elementary properties of natural numbers and integers (e.g., the definition of what a prime number is) and a little overview, to be followed up upon in later chapters, of Diophantine equations, the chord method, Pythagorean triples and Gaussian integers. In the second chapter, he introduces the Euclidean algorithm for finding the greatest common denominator of a pair of numbers, proves unique prime factorization and then launches into an interesting discussion of a graphical map of relatively prime pairs or primitive vectors. Chapter three follows with the arithmetic of congruences, Fermat’s little theorem, Euler’s theorem and the concept of a primitive root. With this, he has assembled the tools for an intelligent explanation of RSA cryptography in chapter four. The reader will enjoy being able to understand this topic, so relevant to real-world applications, and to work out a few toy examples of the encryption/decryption algorithm (with unrealistically small primes, needless to say).
The next chapter, chapter five, takes up the Pell equation, Brahmagupta’s (Indian, seventh century) composition rule for obtaining new solutions from given ones and a general theory of quadratic forms due to Conway. The Pell equation is connected with extensions of the ring of integers of the form Z[√n]. This motivates the case of Gaussian integers (i.e., Z[i]), to which considerable attention is devoted in chapter six. The concept of a Gaussian prime is very interesting and related to unique prime factorization, which continues to hold for Gaussian integers, as Stillwell covers exhaustively.
The adventure gets to be exciting when one takes on other quadratic integers, in chapter seven. As one sees there, unique prime factorization holds for Z[√-2] and Z[ζ_3] but fails for Z[√-3]! One is led by this route to the analysis of algebraic factorizations of polynomial forms such as x^2+y^2 and x^3+y^3. Stillwell gives here an explicit proof of Fermat’s last theorem in the case n=3, what is perhaps the most technically difficult result in the text but not impossible to follow, with the machinery he has built up up to this point.
The next two chapters could be seen as a detour further illustrating pleasant stopping-places in number theory. Chapter eight is given over to Hurwitz’s proof using quaternions that every natural number can be expressed as the sum of four integer squares. Chapter nine goes into the law of quadratic reciprocity, Euler’s criterion and the Chinese remainder theorem (originally from the third to fourth century in China, but dressed up here in its most modern group-theoretical guise).
The last three chapters form the heart of the book and this reviewer’s motivation for reading it in the first place. All the phenomena discussed so far are summed up in the single abstract algebraic concept of a ring. After giving the ring-theoretic axioms and relating rings to their corresponding fields, Stillwell quickly specializes to the case of quadratic fields and their integers. He then pursues the concept of an ideal in a ring, certain subsets that display behavior analogous to a set of multiples of a given number and therefore may be regarded as ‘ideal numbers’, which is how Kummer first conceived of them and how Dedekind formalized the notion. Ideals formed a central part of the modern formulation of abstract algebra at the hands of Emmy Noether during the 1910’s and 1920’s, as presented in van der Waerden’s classic monograph Moderne Algebra. Once the concept has been introduced, the properties of ideals can be derived. Stillwell discusses some cases of principal ideal domains and criteria therefor, and gives as well an example of a non-principal ideal in Z[√-3]. This reviewer remembers a glaring omission in his undergraduate course on algebra: principal ideal domains were discussed without giving a single instance of how an ideal could fail to be principal! The last topic Stillwell covers in chapters eleven and twelve are multiplication of ideals, prime ideals and prime ideal factorization, in which uniqueness can be recovered at the ideal-theoretic level despite the fact that it fails at the number-theoretic level.
In all, Stillwell manages to provide a tour through a significant amount of territory in basic number theory. His pedagogical style ensures that everything he does is comparatively easy to follow and understand. What makes this textbook precious, though, are Stillwell’s commentary along the way and closing section to each chapter, in which he recounts the history of the subject in such a way as to point out just why the concepts he introduces are important. This involves a certain amount of claims about material too advanced to go into in the text itself, but these supply perspective (for instance, on how Kummer was led to conceive of his ideal numbers during the course of a faulty attempt to prove Fermat’s last theorem). The student will come away with enough of a grounding in the simple cases Stillwell does cover in his text to be startled by the amazing properties of the natural numbers and number fields that can indeed be shown to hold in a more advanced treatment of number theory, and to have some clue of the tools and strategy of proof that will be employed.
Stillwell includes some four hundred homework exercises to solidify the reader’s grasp of the material. This reviewer finds most, indeed almost all, of them not to be all that difficult, but perhaps this is a function of his approaching this undergraduate text only after first having gained a fair degree of mathematical maturity in other fields. It may be that an undergraduate venturesome enough to attempt it would find the problems to be for the most part at about the right level to engage his interest. Certainly, they are not all routine and do test one’s understanding of the subject, although Stillwell does throw in a few purely computational problems for good measure.
In view of Stillwell’s masterly pedagogical style, this textbook must be accounted a happy discovery! It could almost—but not quite—convert the reviewer into a number theorist!
Displaying 1 of 1 review