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Elements of Algebra: Geometry, Numbers, Equations
Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.
196 pages, Hardcover
First published August 19, 1994
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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
Everyone knows the romantic legend of the dashing and brilliant Évariste Galois, who founded group theory in 1832 at the age of twenty, only to die shortly thereafter in a senseless duel. For a good historical study of the larger cultural movement in the mathematical community within which the legend takes on its significance, see Amir Alexander’s Duel at dawn: Heroes, martyrs and the rise of modern mathematics. During the course of the nineteenth century, the mathematician’s self-image underwent a sea-change from what it had been in the eighteenth century—the worldly-wise calculator, rooted in physics and a staid Enlightenment rationality—to what it has become ever since, under the influence of Romanticism: the isolated genius, struggling to bring forth his ideal creations against an uncomprehending and hostile world.
Galois was led to his epochal discovery by the study of the solutions of polynomial equations and their symmetries, what was the hot topic of his day. Abel had just shown that the quintic does not admit of a general solution in radicals, as do the quadratic (known to the Babylonians and to every tired high school student), the cubic (discovered by del Ferro around 1500) and the quartic (found by Ferrari, published in Cardano’s Ars Magna of 1545). Galois probed much more deeply than Abel into the systematic properties of the roots of a polynomial over the rationals, their symmetries in particular: the roots map into one another under the action of the Galois group, and there is far more to say than just this about the elegant algebra underlying the problem (vide the fundamental structural theorem of Galois theory with which the last chapter culminates).
John Stillwell undertakes to present elementary algebra at the undergraduate level in the textbook here under review, through to its applications in Galois’ theory of solvability of polynomial equations. As one has come to expect from him, the pedagogical value of his contribution is nothing short of excellent. One notices this right away; the first chapter kicks off with an explanation of straightedge and compass constructions of addition, subtraction, multiplication and extraction of the square root, with an emphasis on what it means for a number to be constructible. One often hears the claim, without substantiation, that one can use the theory of number fields and their extensions to make inferences about the possibility of carrying out geometrical constructions with the ruler and compass; it is nice to see the reason why spelled out here so lucidly by Stillwell.
The pace of this Elements of Algebra is considerably more rapid than the leisurely Elements of Number Theory by the same author. In chapter two, we learn about the natural numbers, integers, rationals, the Euclidean algorithm, unique prime factorization, congruences, Fermat’s little theorem, Euler’s phi function and theorem and the natural connection between the phi function and fractions for two relatively prime numbers.
After a necessary but not too onerous detour in chapter three on irrational and complex numbers and the fundamental theorem of algebra (that every polynomial over the complex field admits at least one root), Stillwell gets down to business in chapter four on polynomials and their factorization resp. irreducibility. Polynomials share with the integers the useful property of division with remainder. The reader is not left high and dry with these concepts, though, as the Eisenstein criterion provides a means of testing for irreducibility (which does not always apply, but which is, practically speaking, the only effective way to show irreducibility). Stillwell illustrates the techniques in two sections on cyclotomic polynomials and their irreducibility, thus killing two birds with one stone, as cyclotomic polynomials are an important topic in their own right.
In preparation for Galois theory, chapter five introduces some elementary material on field extension (through to the Dedekind product theorem, which characterizes iterated extensions). Lest the reader get the impression that the new concepts are otiose, applies Dedekind’s theorem to yield three corollaries in rapid succession on time-honored problems in constructive plane geometry, namely, to show that the problems of duplication of the cube and trisection of the angle are unsolvable and the regular polygons with prime number of sides are constructible only if p-1 is a power of two. From this, it is a quick step to get a necessary condition for the constructibility of the regular n-gon for arbitrary n.
The other half of the preparatory prerequisites for Galois theory fall into place in chapters six and seven on ring and field isomorphism and their connection with symmetric functions, on the one hand, and elementary definition and examples of small finite groups, on the other. Then, in chapter eight, we arrive at last at the crux of the book with the concept of the Galois group of a number field over a subfield, which is nothing but the collection of automorphisms that hold the subfield fixed. By now, we have found the Galois groups for several fields via elementary methods, e.g., Q(√2), the simplest conceivable example. The substance of the chapter lies in an account of extensions by radicals and the non-existence of the general solution by radicals for polynomials of quintic or higher degree. Then Stillwell addresses a topic that does not seem to be very well appreciated, at least in the popular literature: even if the general solution by radicals does not exist, a certain given polynomial of quintic or higher degree could very well admit a solution by radicals, nonetheless. Hence, it takes some work to find specific instances of unsolvable quintic equations.
Chapter nine forms the apex of Stillwell’s treatment of Galois theory, with the theorem of the primitive element and the two fundamental theorems of Galois theory, which involve the concepts of fixed fields, conjugate intermediate fields and normal extensions. Purists will exclaim that having recourse to the theorem of the primitive element sullies the immaculate perfection of Galois theory, which can indeed be formulated without it, but who’s counting here? At our lowly level, all we know are number fields anyway. Perhaps when we are superannuated graduate students we can return to the subject and prove everything without using primitive elements! But Stillwell is wise to limit himself to number fields, as they form the examples most easily accessible to the beginner. Stillwell closes out with two simple applications, namely, to derive necessary and sufficient conditions for the constructibility of the p-gon and for constructible division of arbitrary angles.
Stillwell’s prose has the virtue of being able to convey difficult ideas seemingly effortlessly. His choice of topics is quite suitable for the intended audience and illustrates the material nicely. The reader will be pleased to find, for instance, that he has the tools at hand with which to construct the icosahedron or to prove irreducibility of a given polynomial with the aid of Eisenstein’s criterion or a simple variant on it. The exposition is enhanced incomparably by the historical commentary, given largely in the thorough chapter-end notes. To see how what one is learning fits into the larger narrative of mathematical progress is invaluable in motivating the reader to press on. If this were not enough, the thoughtfully chosen homework exercises will encourage him. Comparatively few are of the mechanical plug-in kind, but none is so hard as to discourage the student. Opinions diverge at this point; some will aver the student needs to be tested beyond the limit of his ability (nightmare-inducing memories of Herstein in college crop up in this connection), but let us be content that Stillwell’s problems do, at least, force the student to think a little and to review the material just covered. As such, this text is appropriate for those who are learning on their own without anyone with whom to confer when stuck. They must merely bear in mind that they will not have mastered modern algebra in all its glory upon completing the text but will only have been exposed to its elements, as the title says. While Stillwell does achieve a certain pleasing degree of completeness in rounding out the subject of Galois theory (limited to number fields, of course), his text is by no means comprehensive enough to constitute an adequate entryway into the field of algebra as a whole at the graduate level. One will have to go on to Dummit and Foote, Artin or Hungerford (or possibly a newer competitor such as Aluffi), which bridge the gap between the advanced undergraduate and graduate levels (and, as is known, feelings run high as to which author or authors at this stage are best; certainly, though, it would be crazy to start with Lang). By the time one gets to this level, though, one might miss something that Stillwell accomplishes quite well, viz., to keep an eye on the roots of the algebra in geometrical problems readily accessible to visual intuition, which, after all guided the originators of the subject up to and all through the nineteenth century. Many small finite groups, for instance, have realizations as the set of symmetries of simple geometrical figures; the group of rotational symmetries of the tetrahedron is isomorphic to the alternating group of order 4 etc. Twentieth-century abstract algebra, however, has almost entirely shed any trace of a connection to spatial intuition, as in van der Waerden’s trendsetting Moderne Algebra of 1931 (the style of which Bourbaki was certainly to pick up on).
Galois was led to his epochal discovery by the study of the solutions of polynomial equations and their symmetries, what was the hot topic of his day. Abel had just shown that the quintic does not admit of a general solution in radicals, as do the quadratic (known to the Babylonians and to every tired high school student), the cubic (discovered by del Ferro around 1500) and the quartic (found by Ferrari, published in Cardano’s Ars Magna of 1545). Galois probed much more deeply than Abel into the systematic properties of the roots of a polynomial over the rationals, their symmetries in particular: the roots map into one another under the action of the Galois group, and there is far more to say than just this about the elegant algebra underlying the problem (vide the fundamental structural theorem of Galois theory with which the last chapter culminates).
John Stillwell undertakes to present elementary algebra at the undergraduate level in the textbook here under review, through to its applications in Galois’ theory of solvability of polynomial equations. As one has come to expect from him, the pedagogical value of his contribution is nothing short of excellent. One notices this right away; the first chapter kicks off with an explanation of straightedge and compass constructions of addition, subtraction, multiplication and extraction of the square root, with an emphasis on what it means for a number to be constructible. One often hears the claim, without substantiation, that one can use the theory of number fields and their extensions to make inferences about the possibility of carrying out geometrical constructions with the ruler and compass; it is nice to see the reason why spelled out here so lucidly by Stillwell.
The pace of this Elements of Algebra is considerably more rapid than the leisurely Elements of Number Theory by the same author. In chapter two, we learn about the natural numbers, integers, rationals, the Euclidean algorithm, unique prime factorization, congruences, Fermat’s little theorem, Euler’s phi function and theorem and the natural connection between the phi function and fractions for two relatively prime numbers.
After a necessary but not too onerous detour in chapter three on irrational and complex numbers and the fundamental theorem of algebra (that every polynomial over the complex field admits at least one root), Stillwell gets down to business in chapter four on polynomials and their factorization resp. irreducibility. Polynomials share with the integers the useful property of division with remainder. The reader is not left high and dry with these concepts, though, as the Eisenstein criterion provides a means of testing for irreducibility (which does not always apply, but which is, practically speaking, the only effective way to show irreducibility). Stillwell illustrates the techniques in two sections on cyclotomic polynomials and their irreducibility, thus killing two birds with one stone, as cyclotomic polynomials are an important topic in their own right.
In preparation for Galois theory, chapter five introduces some elementary material on field extension (through to the Dedekind product theorem, which characterizes iterated extensions). Lest the reader get the impression that the new concepts are otiose, applies Dedekind’s theorem to yield three corollaries in rapid succession on time-honored problems in constructive plane geometry, namely, to show that the problems of duplication of the cube and trisection of the angle are unsolvable and the regular polygons with prime number of sides are constructible only if p-1 is a power of two. From this, it is a quick step to get a necessary condition for the constructibility of the regular n-gon for arbitrary n.
The other half of the preparatory prerequisites for Galois theory fall into place in chapters six and seven on ring and field isomorphism and their connection with symmetric functions, on the one hand, and elementary definition and examples of small finite groups, on the other. Then, in chapter eight, we arrive at last at the crux of the book with the concept of the Galois group of a number field over a subfield, which is nothing but the collection of automorphisms that hold the subfield fixed. By now, we have found the Galois groups for several fields via elementary methods, e.g., Q(√2), the simplest conceivable example. The substance of the chapter lies in an account of extensions by radicals and the non-existence of the general solution by radicals for polynomials of quintic or higher degree. Then Stillwell addresses a topic that does not seem to be very well appreciated, at least in the popular literature: even if the general solution by radicals does not exist, a certain given polynomial of quintic or higher degree could very well admit a solution by radicals, nonetheless. Hence, it takes some work to find specific instances of unsolvable quintic equations.
Chapter nine forms the apex of Stillwell’s treatment of Galois theory, with the theorem of the primitive element and the two fundamental theorems of Galois theory, which involve the concepts of fixed fields, conjugate intermediate fields and normal extensions. Purists will exclaim that having recourse to the theorem of the primitive element sullies the immaculate perfection of Galois theory, which can indeed be formulated without it, but who’s counting here? At our lowly level, all we know are number fields anyway. Perhaps when we are superannuated graduate students we can return to the subject and prove everything without using primitive elements! But Stillwell is wise to limit himself to number fields, as they form the examples most easily accessible to the beginner. Stillwell closes out with two simple applications, namely, to derive necessary and sufficient conditions for the constructibility of the p-gon and for constructible division of arbitrary angles.
Stillwell’s prose has the virtue of being able to convey difficult ideas seemingly effortlessly. His choice of topics is quite suitable for the intended audience and illustrates the material nicely. The reader will be pleased to find, for instance, that he has the tools at hand with which to construct the icosahedron or to prove irreducibility of a given polynomial with the aid of Eisenstein’s criterion or a simple variant on it. The exposition is enhanced incomparably by the historical commentary, given largely in the thorough chapter-end notes. To see how what one is learning fits into the larger narrative of mathematical progress is invaluable in motivating the reader to press on. If this were not enough, the thoughtfully chosen homework exercises will encourage him. Comparatively few are of the mechanical plug-in kind, but none is so hard as to discourage the student. Opinions diverge at this point; some will aver the student needs to be tested beyond the limit of his ability (nightmare-inducing memories of Herstein in college crop up in this connection), but let us be content that Stillwell’s problems do, at least, force the student to think a little and to review the material just covered. As such, this text is appropriate for those who are learning on their own without anyone with whom to confer when stuck. They must merely bear in mind that they will not have mastered modern algebra in all its glory upon completing the text but will only have been exposed to its elements, as the title says. While Stillwell does achieve a certain pleasing degree of completeness in rounding out the subject of Galois theory (limited to number fields, of course), his text is by no means comprehensive enough to constitute an adequate entryway into the field of algebra as a whole at the graduate level. One will have to go on to Dummit and Foote, Artin or Hungerford (or possibly a newer competitor such as Aluffi), which bridge the gap between the advanced undergraduate and graduate levels (and, as is known, feelings run high as to which author or authors at this stage are best; certainly, though, it would be crazy to start with Lang). By the time one gets to this level, though, one might miss something that Stillwell accomplishes quite well, viz., to keep an eye on the roots of the algebra in geometrical problems readily accessible to visual intuition, which, after all guided the originators of the subject up to and all through the nineteenth century. Many small finite groups, for instance, have realizations as the set of symmetries of simple geometrical figures; the group of rotational symmetries of the tetrahedron is isomorphic to the alternating group of order 4 etc. Twentieth-century abstract algebra, however, has almost entirely shed any trace of a connection to spatial intuition, as in van der Waerden’s trendsetting Moderne Algebra of 1931 (the style of which Bourbaki was certainly to pick up on).
June 3, 2018
Well written and I needed the background!
Displaying 1 - 2 of 2 reviews