Review of The Best Writing on Mathematics 2012


In describing the essays in this volume as the “best” writing on mathematics, the word “best” can’t be taken literally. For one thing a mathematician would naturally point out that there is no simple, obvious linear ordering on the set of writings about mathematics that reflects “quality”. There is certainly no metric to quantify quality for this type (or any other type) of writing. Specifying a particular audience that might have useful opinions on quality would help, but still be inadequate. That said, let’s assume the audience is at least people who understand some mathematics and who appreciate it and value it. That’s still a number of audiences, since it includes professional mathematicians, teachers of mathematics at all levels, and users of mathematics in fields like physics, statistics, economics, etc.

There’s a little something in this volume for many of these audiences, but probably not enough to satisfy most of them. The editor of this collection is a specialist in mathematics education, so that category is over-represented. People who apply mathematics in their own special fields of expertise as well as research mathematicians will probably find the material on mathematics education to be of minor interest.

Many of the other topics tend to be treated in a way that would appeal to that fairly small part of the “general public” that actually has any interest at all in mathematics. Check out the table of contents to see if any topic of that sort interests you. Since this is a personal evaluation, I’ll just mention articles that seemed interesting to me.

Perhaps unsurprisingly, the two best articles in the volume were written by research mathematicians who’ve won Fields Medals: Terrence Tao and Timothy Gowers. Tao writes about a topic he knows extremely well (of which there are quite a few): the distribution of prime numbers. The question is whether there are any patterns in how prime numbers are distributed, or whether the distribution is essentially random. Gowers has two contributions, but the interesting one is the question of whether mathematical truths are “invented” or “discovered”. This can be treated as either an empirical question or a philosophical one. Gowers focuses on the empirical question, and the answer is “both”.

There are several good articles on the history of mathematics. One (by Peter Rowlett) offers a number of examples where solutions to practical problems were enabled by theoretical work that may have occurred centuries earlier. An article by John Baez and John Huerta discusses unfamiliar sorts of “numbers”, including quaternions and octonions, and how they are relevant to symmetry, especially in string theory (as in theoretical physics). Charlotte Simmons writes about the influence of the logician Augustus De Morgan on other notables, such as William Hamilton (discoverer of quaternions) and George Boole (inventor/discoverer of Boolean algebra). Fernando Gouvea shows how the modern theory of real numbers and of sets arose from interactions over many years between George Cantor and his teacher, Richard Dedekind. A satisfactory theory was discovered only after various promising but false starts.

Only three other articles seem worthy of mention. Brian Hayes explains the remarkable topological fact that the volume of a unit n-ball (n-dimensional solid sphere of radius 1) approaches 0 as n tends to infinity. Richard Elwes writes about the very esoteric theory of infinite cardinal numbers (as originally developed by George Cantor). Some of the most recent work has been done by Hugh Woodin, and it suggests that even Kurt Gödel and Paul Cohen didn’t have the last word on Cantor’s Continuum Hypothesis.

Lastly, philosopher Ian Hacking asks the question, “Why is there a philosophy of mathematics at all?” As a philosopher, he attempts to present several plausible justifications. This discussion concerns mathematical Platonism and the “invented” vs. “discovered” issue. Professional mathematicians, though, aren’t that concerned with the philosophy, since for them the really important thing is proving theorems, while letting philosophers worry about what that “means”.

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Review of The Mathematician’s Brain

Birth of a Theorem

The title is slightly misleading, as it might lead one to expect an analysis of how mathematicians work based on psychology or neuroscience. In fact, there is very little of that, especially if one discounts a short chapter on Freud’s views (which don’t really have much of value on the subject). So it would be justifiable to be disappointed with the book if one had that expectation.

While it would be great if someday scientific study of the brain could reveal the mechanisms of creativity in mathematics or in general, that’s not the situation now. Ruelle does report the ideas on mathematical creativity of Poincaré and Hadamard, but there hasn’t been much to add in the last 100 years. Nevertheless, the present book is an excellent follow-on to Ruelle’s Chance and Chaos (reviewed here). These two have much in common: they are relatively short, comprise a number of brief chapters on diverse topics, and are quick reads. Both books cover a lot more ground than their titles would suggest. But, happily, there’s little overlap between them.

In one respect the present book is even shorter than its 130 pages (exclusive of the excellent notes at the end) would suggest, because it’s really two books in one. The first part, about 60 pages, is mostly nontechnical and quite accessible to general readers. It gives, in that short space, a clear picture of the general subject matter of concern to mathematicians since the ancient Greeks. (Which is far more than just arithmetic, geometry, algebra, and calculus.) The remainder of the book deals with more technical topics. There are some more detailed explanations in the notes. However, a reader without some college-level math courses would need to be satisfied to accept the technical terminology without much explication.

But the reward from reading the later chapters, for readers with any amount of mathematical background, is a clear picture of how professional mathematicians actually work. I’ve given the book a top rating for its coverage of topics such as sketches of one of the 20th century’s most original mathematicians, Alexander Grothendieck (whom Ruelle knew personally), set theory as a foundation for mathematics, Gödel’s incompleteness theorem (very succinctly explained), and the Riemann Hypothesis (and the intriguing idea that if RH could be proven to be undecidable within standard arithmetic, it should be provable in a broader theory).

There’s a lot more. The general idea of mathematical structures is explained, and how new concepts are created. This leads to a discussion of whether mathematical concepts are “created” or “discovered” – a very old debate. (There seems to be consensus now that the truth is “some of both”.) There’s a discussion of a theorem, the “Circle Theorem”, which was actually discovered by physicists T. D. Lee and C. N. Yang (best known as winners of a Nobel Prize for discovering CP parity nonconservation). The Circle Theorem, which isn’t well-known to mathematicians, is surprising but simple to state, and has a relatively simple proof.

Finally, there is a discussion of the nitty-gritty details of proving some of the complicated theorems of modern mathematics. In general, what makes this activity difficult is the huge number of possible approaches to finding proofs. It’s like finding one’s way through a high-dimensional labyrinth to a valid proof. There are a huge number of possible paths to explore, almost all of which lead nowhere. Yet, somehow, a process of enlightened “tinkering” with the details of a proof can succeed. This involves proving just the right set of lemmas with a variety of slightly different hypotheses and conclusions in order to reach the desired result. The chapter on “The Strategy of Mathematical Invention” gives a sketch of how this is done, but there are no explicit examples. Fortunately, for anyone who’s curious about this, Cédric Villani’s Birth of a Theorem (reviewed here) provides excruciating details of one example of this process.

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Review of My Brain Is Open


It’s probably fair to say that a large majority of the general public in the U. S. could not name a single important mathematician who was active in the period 1933 to 1996 (the years of Paul Erdős’ adult life). A few people might think of Andrew Wiles, John von Neumann, or perhaps Kurt Gödel. It would be surprising if more than a percent or two could mention Paul Erdős or even recognize the name. Yet Erdős is probably one of the dozen or so most important mathematicians of that period. For instance, nobody in the history of the world published more new mathematical results (almost 1500), except for Leonhard Euler (who died in 1783).

Erdős created himself or was among the creators of a number of new mathematical specialties, including combinatorics, Ramsey theory, probabalistic number theory, and combinatorial geometry. He also made ground-breaking advances in the early stages of set theory and graph theory. And he added very substantially to classical number theory (one of Euler’s specialties). In addition to his astonishing productivity, he remained active and prolific until he died at the age of 83 – something almost unheard of among mathematicians. Nevertheless he had little or no interest in many other branches of modern mathematics, such as topology, abstract algebra, or mathematical physics.

Schechter’s book is very good at explaining in general terms what each of those mathematical topics are about and the contribution made by Erdős. Unlike most modern mathematics, the things that interested Erdős the most were “simple” things, like numbers and geometrical figures that are familiar to most people. So it’s easy enough to understand most of what Erdős worked on. It would be nearly impossible, however, to explain to non-mathematicians the techniques by which the results were obtained. (After all, his results had eluded all earlier mathematicians.) And even mathematicians are mystified by the mental processes that led to the results.

Erdős was quite an unusual person in other respects as well. He never married or had any apparent sexual interests. During most of his adult life he had no permanent home and almost no possessions other than a couple of suitcases, some notebooks, and a few changes of clothes. He was almost perpetually on the go from one place to another after his welcome at one host’s abode started to wear thin. And his memory was phenomenal – though only for details that were important to him, namely anything relevant to mathematics he cared about and his vast network of mathematical collaborators.

He co-authored papers with almost 500 other mathematicians, and personally discussed mathematics with hundreds of others. He generally knew the phone numbers and other personal details of most in his network. (Yet he was often unable to associate their names and details with their faces when he encountered them at meetings.) He was also able to recall technical details and publication information of thousands of mathematical papers, which may have been published decades previously. This fact probably helps explain his ability in many cases to solve new problems within minutes, because he could recall such a vast number of earlier results and techniques.

Schechter’s biography is generally quite good. The author has a PhD in physics and is obviously conversant with the mathematics that interested his subject. There are just a few minor issues. Although Schechter never actually met Erdős, many personal anecdotes are reported. Those must have come from conversations with associates of Erdős or articles about him, but there’s only a two-page “Note on Sources” instead of footnotes with specific details. There is a good bibliography, however.

The book is only about 200 pages, so it’s a quick read. However, if it had been a little longer, it could have gone into somewhat more detail about Erdős’ mathematics – such as set theory and probability theory. Another Erdős biography (The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth) was published about the same time, less than 2 years after the subject’s death. (I’ve reviewed that one too.) So both biographers most likely wanted to get published as quickly as possible. There are clear signs of haste in both cases. In the present book there are few things that a proofreading should have caught, but only one more serious error: in Chapter 4 there are incorrect references to two of the graph diagrams.

All in all, this biography can be highly recommended for an overview of Erdős’ mathematics, a fine portrait of a very unique and colorful individual, and the opportunity to gain a little understanding of the social process in which mathematics of the highest caliber is actually created.

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Review of The Man Who Loved Only Numbers


When Paul Erdős died in 1996 Paul Hoffman had known him for about 10 years and interviewed him a number of times. Hoffman had also known and interviewed many of Erdős’ friends and associates. So it’s fair to say that Hoffman had a lot more knowledge of the subject of his biography than most biographers (unless they’re family or close friends) ever do. The biography is indeed quite good, and provides a clear and informative portrait of the very unique, appealing, and colorful individual that Erdős was.

However, what defined Paul Erdős even more than his idiosyncrasies and his humanity was that he was quintessentially a mathematician – a singularly talented and prolific one. Another biography of Erdős, by PhD physicist Bruce Schechter (My Brain is Open: The Mathematical Journeys of Paul Erdos), was published at almost the same time as Hoffman’s. (I have reviewed that as well, and I’ll try not to duplicate much from that other review.) Hoffman freely concedes that he is not a mathematician. While Schechter isn’t a professional mathematician, he is more able to give a complete and accurate account of Erdős’ work than Hoffman.

The picture of Erdős that comes across in both biographies is much the same, since both books are based on a lot of the same source material. Hoffman’s book has the advantage that the source of all Erdős quotes (except those Hoffman got directly) are documented in footnotes. But because Hoffman also can report many anecdotes from his interviews with Erdős himself and with his associates, this biography offers a somewhat clearer picture of the subject as a person, apart from his work.

One anecdote is especially revealing. On an occasion in the late 1960s Erdős (who was about 55 at the time) was staying with an old friend from Hungary in southern California, very near the beach. One day Erdős went out to walk on an esplanade above the beach. Only ten minutes later his hosts received a phone call, from someone who lived close to the esplanade about five blocks away, to report that Erdős turned up on their doorstep saying he was lost and needed help finding his way back to his friend’s place. So Erdős, in spite of his prodigious memory for details of his mathematics and his mathematical collaborators, couldn’t even remember how to retrace his steps. The natural explanation is that he was so absorbed in his mathematical thoughts that no vestige of his short walk had registered in his memory.

As other anecdotes made clear, Erdős was fully capable of recalling details of mathematical conversations he’d engaged in years before, and he could also keep track of more or less simultaneous conversations he carried on with several different mathematicians at the same meeting. He could also recall details of perhaps thousands of technical papers he’d read decades before. It seems reasonable to conclude that his ability to concentrate and to recall mathematical detail had a great deal to do with his singular power as a mathematician – as exemplified by his ability either to solve quickly new mathematical problems or at least to judge accurately their level of difficulty almost effortlessly.

In contrast to this clear portrait of Erdős as a person, Hoffman’s lack of mathematical background means he must rely on the testimony of others he interviewed to describe Erdős’ mathematics. The result is a somewhat less satisfactory account. One example is what Hoffman calls “friendly numbers”. He says this means a pair of numbers (a and b) where the sum of proper divisors of a is equal to b and the sum of proper divisors of b is equal to a. This is actually the definition of “amicable numbers”. The example given is the pair 220 and 284, which was known to Pythagoras. Those are indeed “amicable numbers”. The accepted definition of “friendly numbers” is numbers that have the same ratio between themselves and their own sum of divisors. By this definition, 220 and 284 aren’t “friendly”.

An even more serious problem is the discussion of Bernhard Riemann’s non-Euclidean geometry. Hoffman writes “He [Riemann] builds a seemingly ridiculous assumption that it’s not possible to draw two lines parallel to each other. His non-Euclidean geometry replaces Euclid’s plane with a bizarre abstraction called curved space.” It’s not actually bizarre at all, since the surface of any sphere is one example. Straight lines on the surface of a sphere are (parts of) “great circles” (which are by definition the largest circles that can be drawn on a sphere, like the Earth’s equator). Great circles are never “parallel”, since they always intersect.

Apart from these and a few other mathematical glitches, Hoffman’s book is almost free of trivial proofreading errors. But there’s one glaring exception (at least in the paperback edition). Sixteen pages of pictures are included in the middle of the book, and given appropriate page numbers. However, the footnotes at the end of the book are all keyed to page numbers, and they haven’t been corrected to account for the picture pages, so that all references to pages after the pictures are off by 16 – a very annoying problem if one wants to actually check these references.

In spite of these problems, Hoffman’s book provides a fine portrait, based on personal experiences, of Erdős the man.

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Review of Mathematics without Apologies


Michael Harris’ book is definitely one that mathematicians shouldn’t miss. It’s also a must-read for people who aren’t professional mathematicians but do have a deep appreciation for the subject and have more than a passing interest in understanding what mathematics is “about”. It will not, however, win any converts among people who think they “don’t like math” and would require serious persuasion to change their minds.

As the title should make clear, the book is intended as a counterpoint to G. H. Hardy’s A Mathematician’s Apology. Hardy poses the question “[W]hy is it really worthwhile to make a serious study of mathematics? What is the proper justification of a mathematician’s life?” Hardy’s answer is somewhat defensive and hedged. He rejects a justification based on practical usefulness of mathematics, which he doesn’t defend robustly. Instead, he concludes that the justification for the life of any “real” mathematician is to “have added something to knowledge, and have helped others to add more; and that these somethings have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any other artists, great or small, who have left some kind of memorial behind them.”

Harris, on the other hand, rejects a defensive stance and aims to justify mathematics and the work of mathematicians “without apology”. He stands with Hardy in likening mathematics to other fine arts. But he goes further to justify both mathematics and other art forms not in utilitarian or moralistic terms, but in aesthetic terms of the pleasure from creating and appreciating art of high quality. His argument is first set out in Chapter 3 (“Not Merely Good, True, and Beautiful”). And after excursions through diverse mathematical, philosophical, biographical, artistic, and literary topics, the argument concludes in Chapter 10 (“No apology”). Along the way, it’s noted that other top tier scientists besides mathematicians have also justified their work in terms of pleasure. There is, for instance, Richard Feynman, one of whose books is The Pleasure Of Finding Things Out. Although the things studied in physics and mathematics are rather different (though some like Max Tegmark (Our Mathematical Universe) disagree), there is a distinct pleasure associated with important discoveries in any scientific field. Creative art also involves discovery and “finding things out”. Both the creative artist/scientist and others who simply learn to appreciate their work experience the associated pleasure.

Michael Harris’ most noteworthy mathematical accomplishment is his work within a vast generalization of number theory (in more classical forms of which Hardy also made his mark many decades ago). This work is part of an extensive series of related conjectures called “The Langlands Program”, after Robert Langlands who conceived it and has contributed to its continual expansion. The LP is highly abstruse, technical mathematics, and Harris makes no serious attempt in this book to sketch its outlines. He touches only briefly on some of the more concrete manifestations of the program in connection with what are known as “elliptic curves”. A very concise way to describe the LP is in terms of elaborate “correspondences” between seemingly unrelated mathematical objects such as “Galois representations” and “automorphic forms”. There’s a more complete semi-technical discussion of these correspondences in Edward Frenkel’s Love and Math, and a technical introduction (accessible only to math graduate students and professionals) is An Introduction to the Langlands Program.

Some people have found Harris’ book frustrating because it seems to lack “structure”. In fact, however, it does have structure, but as is often true in advanced mathematics, the structure isn’t always readily apparent. As Harris would point out, that is precisely what appeals to mathematicians about their vocation and avocation: its pleasure lies in discovering hidden structures that aren’t obvious on the surface. The Langlands Program is an apt metaphor for mathematics itself. Earlier examples of the discovery of rich hidden structures in mathematics abound – from Galois theory to Hilbert spaces.

Many topics are discussed in the book, some only cursorily, some in more depth. Here are just a few of them; mathematical Platonism (Harris, like most mathematicians, is an adherent), category theory, Hindu and Buddhist philosophy, literature (such as Thomas Pynchon’s), mathematical tricks, the mind-body problem, economics and finance, mathematical “charisma”, life in Paris, mathematics and sex (in reference to Frenkel’s Love and Math – and much more. Some of this is autobiographical, but there’s almost always some connection, anecdotal or otherwise, with the book’s main themes. It’s a smörgåsbord.

The book definitely isn’t light reading. It’s also heavily footnoted – an average of about 60 per chapter. (Most are worth reading.) There’s also a running series of imagined dialogues – at a very elementary level – on “How to Explain Number Theory at at Dinner Party”. So don’t expect to finish it in a day or two. A better approach is to take it a chapter at a time, and allow that to sink in before proceeding.

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Algebraic number theory – index

This is a list of posts in the algebraic number theory series to date, oldest first.

1. A brief history of algebra

2. Numbers – rational and irrational, real and imaginary

3. Diophantine equations

4. Groups and rings

5. Fields and Galois theory

6. Modular arithmetic

7. Rings of algebraic integers

8. Rings and ideals

9. Uniqueness of factorization

10. Failure of unique factorization

11. Why do we care about unique factorization?

12. More concepts from ring theory

13. Factorization of prime ideals in extension fields

14. Splitting of prime ideals in quadratic extensions of ℚ, part 1

15. Splitting of prime ideals in algebraic number fields

16. Roots of unity and cyclotomic fields

17. Cyclotomic fields, part 2



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Cyclotomic fields, part 2

In our previous article on cyclotomic fields we were talking about why the Galois group G of ℚ(μn)/ℚ is isomorphic to (ℤ/nℤ)×, where n∈ℤ and μn is the group of nth roots of unity, the roots of xn-1=0 in some extension of ℚ. (Check here for a list of previous articles on algebraic number theory.)

So far we’ve shown that G is isomorphic to a subgroup of (ℤ/nℤ)×. We still need to show it is actually isomorphic to the whole group, or equivalently that |G|, the order of G, which is equal to the degree of the field extension, [ℚ(μn):ℚ], actually equals |(ℤ/nℤ)×|, which is φ(n), the number of positive integers less than and relatively prime to n.

The group μn is cyclic. Any generator of the group is, by definition, a primitive nth root of unity. We let ζ be an arbitrary but fixed such generator. Then ℚ(μn)=ℚ(ζ). Let f(x) be the minimal polynomial of ζ. f(x)∈ℚ[x] is irreducible over ℚ. All other elements of μn are of the form ζa for some a∈ℤ, where a is well-defined modulo n. Further, ζa is a primitive nth root of unity if and only if a is relatively prime to n, i. e. the greatest common divisor (a,n)=1. The degree of f(x) equals [ℚ(μn):ℚ] and |G|. At this point, all we know about these numbers is that they divide φ(n) (since |G| does).

The group homomorphism j:G→(ℤ/nℤ)× was defined by the relation σ(ζ)=ζj(σ), for σ∈G. We showed that j(σ) is injective and independent of the choice of ζ. Hence G is isomorphic to a subgroup of (ℤ/nℤ)×, and therefore the degree of f(x) (and [ℚ(μn):ℚ] and |G|) is ≤φ(n). G is isomorphic to a proper subgroup of (ℤ/nℤ)×, i. e. not the whole group, if j is not surjective and the degree of f(x) is strictly less than φ(n).

As we noted last time, the problem here is that we don’t yet know that all φ(n) primitive nth roots of unity are zeroes of f(x), so that |G| and the degree of f(x) equal φ(n), and hence G(ℚ(μn)/ℚ) ≅ (ℤ/nℤ)×. Stated another way, we don’t yet know that the field homomorphism on ℚ(μn) induced by mapping ζ to ζa is actually an automorphism of the field, hence an element of the Galois group. It could fail to be if, say, the minimal polynomial of ζa is different from that of ζ, which could happen if the degree of f(x) is less than φ(n) because not all primitive nth roots of unity are zeroes of f(x). In order to rule out this possibility, we will show that the degree of f(x) is ≥φ(n).

There are various ways to prove the isomorphism, and even a number of ways to prove that f(x) has φ(n) distinct roots, so its degree is ≥φ(n). Many of these proofs use machinery (such as discriminants, factorization and ramification of primes, etc.) that we haven’t extensively discussed yet, so I’ll avoid using such things in the proof. However, we’ll get to these topics eventually, and also show a way to construct the automorphisms σ∈G explicitly – after finishing the proof that G(ℚ(μn)/ℚ) ≅ (ℤ/nℤ)×.

So let’s get started. The roots of the minimal polynomial f(x)∈ℚ[x] are all conjugates σ(ζ) for σ∈G, so f(x)=Πσ∈G(x-σ(ζ)). Hence f(x) is monic (leading coefficient 1). The coefficients of f(x) are symmetric functions of all conjugates of ζ, so the coefficients are all left fixed by all σ∈G. f(x) divides xn-1, so all its roots – the conjugates of ζ – are algebraic integers. So the coefficients are also algebraic integers (sums of products of powers of algebraic integers) – members of the ring of integers Oℚ(ζ). They are also in the base field, since they’re left fixed by G. A basic fact is that Oℚ(ζ)∩ℚ = ℤ – any algebraic integer that lies in the base field is necessarily an integer of the base field. Hence f(x)∈ℤ[x].

Suppose that for any a∈ℤ relatively prime to n, i. e. (a,n)=1, ζa is also a root of f(x): f(ζa)=0. Since these ζa with 1≤a<n are distinct primitive nth roots of unity if ζ is, and there are φ(n) of them, the degree of f(x), and hence |G|, is ≥ φ(n). But we already showed |G|≤φ(n), hence |G|=φ(n). Since G is isomorphic to a subgroup of (ℤ/nℤ)×, we must actually have an isomorphism: G ≅ (ℤ/nℤ)×.

So all we have to show is f(ζa)=0 for 1≤a<n and (a,n)=1. The first thing to note is that it suffices to prove this just for primes p with (p,n)=1. For suppose we had that. For general a with (a,n)=1, let p be a prime that divides a. Then (p,n)=1. Consider ζa/p. Since (a/p,n)=1, ζa/p is a primitive nth root of unity with one fewer prime divisor in the exponent than ζa. So by induction on the number of prime divisors of the exponent f(ζa/p)=0. But if the result is true for prime powers of primitive nth roots of unity that satisfy f(x)=0, then f(ζa)=0 since ζa=(ζa/p)p. Alternatively, you can recall that (according to a theorem of Dirichlet), there are infinitely many primes p in the arithmetic progression a+nk for (a,n)=1 and k∈ℤ. Since ζn=1, ζap for all such p.

So let p be prime and (p,n)=1. Then note that f(x)p-f(xp)∈pℤ[x] for any f(x)∈ℤ[x]. This can be proved by induction on the degree of f(x). Suppose the highest degree term of f(x) is Axm, with p∤A. Then (Axm)p-Axmp∈pℤ[x] because Ap≡A (mod p), because (ℤ/pℤ)× is cyclic of order p-1 (Fermat’s theorem). So if h(x)=f(x)-Axn, then we just have to show h(x)p-h(xp)∈pℤ[x]. But that can be assumed true by induction, unless the degree of h(x) is 1. In the latter case, if h(x)=Ax+B, we need (Ax+B)p-(Axp+B)∈pℤ[x]. But all the coefficients in (Ax+B)p except the first and last contain binomial coefficients divisible by p, and the remaining terms are handled with Fermat’s theorem as before.

Finally, then, suppose the opposite of what we want to show, namely that there is a prime p with p∤n and f(ζp)≠0. By what we just showed, f(ζp) is divisible by p in Oℚ(ζ). We have f(x)=Πi∈I(x-ζi) for I={i∈ℤ: 1≤i<n and (i,n)=1}. So f(ζp) divides a product of nonzero factors ζpi. By a lemma we’ll prove in a moment, if J={(i,j): i,j∈ℤ, 0≤i,j<n, i≠j}, Π(i,j)∈Jij) = (-1)n-1nn. Hence f(ζp) divides nn and p|n, contrary to assumption. This contradiction means f(ζp)=0, as required. We’ve now shown f(x) has at least φ(n) roots, hence G(ℚ(μn)/ℚ) ≅ (ℤ/nℤ)×.

Now for the last lemma: We have xn-1=Π0≤i<n(x-ζi). Equating the constant terms gives (-1)n-10≤i<nζi. And by taking derivatives of both sides, nxn-10≤i<nΠ0≤j<n, j≠i(x-ζj). Substituting x=ζk, nζk(n-1)0≤j<n, j≠kkj). Taking products of this for 0≤k<n gives, with the set J as above, Π(i,j)∈Jij) = nn0≤k<nζk)n-1. But the last product on the right side was evaluated above, so finally we are left with (-1)n-1nn on the right (since n-1 has the same even/odd parity as its square).

Well, that was a bit of work, wasn’t it? But nothing too esoteric, apart from a little Galois theory and some classic number theoretical facts. (Thanks to [1, pp 96-8] for the bulk of the proof.)

Actually, it is possible to do this without the lemma, using the theorem on primes in an arithmetic progression. Suppose f(x) is any polynomial in ℤ[x] such that f(ζ)=0 when ζ is a primitive nth root of unity. Then for any a∈ℤ with (a,n)=1, since f(x)p-f(xp)∈pℤ[x] for any prime p∈ℤ, we have 0=f(ζ)p≡f(ζp) mod pOℚ(ζ). But there are infinitely many primes p≡a mod n, and for such p, ζpa. Consequently, f(ζa) is a member of an infinite number of distinct prime ideals, which is possible only if f(ζa)=0. Hence f(x) has degree ≥φ(n), which is the crucial fact we found before.

We can now define the cyclotomic polynomial Φn(x)=Π0<i<n, (i,n)=1(x-ζi), for any primitive nth root of unity ζ. From the foregoing, we know a lot about Φn(x): its roots are precisely all the primitive nth roots of unity (in ℂ), its degree is φ(n), it is irreducible (over ℚ), its coefficients are in ℤ, and it is the minimal polynomial of ζ. The notation Φn(x) is on account of its relation to the Euler function φ(n).

We also have this factorization of xn-1 in ℤ[x]: xn-1 = Πd|nΦd(x). This holds, since the roots of each Φd(x) are precisely the roots of unity in the cyclic group μn that have exact order d for each d that divides n. (Each root has one and only one exact order d satisfying d|n.) This relation is occasionally useful, and it yields interesting facts such as Σd|nφ(d) = n (by taking degrees of polynomials on both sides).

It turns out that the irreducibility of Φn(x) is relatively easy to prove for certain n, namely those that are powers of a single prime. So let p be prime and q=pr for an integer r≥1. Let f(x)=Φq(x). The roots of f(x) are primitive qth roots of unity, namely ζ∈μq such that ζ has order q. There are φ(q) of these and φ(q)=q-q/p=q(1-1/p)=(p-1)pr-1 (because every pth element of the set {0,1,…,q-1} is divisible by p). So clearly f(x)=(xq-1)/(xq/p-1). Let g(x)=(xp-1)/(x-1) and h(x)=g(x+1)=((x+1)p-1)/x=xp-10<j<p(p j)xj-1, where (p j) is a binomial coefficient, which is divisible by p if 0<j<p. Finally, consider the polynomial h(xq/p)=g(xq/p+1).

Suppose f(x) splits in ℚ[x]. Then since f(x)=g(xq/p), the latter splits, and consequently g(xq/p+1)=h(xq/p) does too. But h(xq/p) is what’s known as an Eisenstein polynomial, because the leading coefficient is not divisible by p, the constant term is p (not divisible by p2), and all other nonzero coefficients (the binomial coefficients) are divisible by p. However, Eisenstein polynomials are irreducible over ℚ. This contradiction means f(x) must be irreducible over ℚ. QED.

The fact that Φq(x) is irreducible if q=pr, and hence G(ℚ(μq)/ℚ)≅(ℤ/qℤ)×, can be used as the basis for yet another proof of this isomorphism for arbitrary n, by considering prime power divisors q of n, the corresponding extensions ℚ(μq)/ℚ, and their Galois groups in building up the full extension ℚ(μn)/ℚ and its Galois group. But we won’t go into that now.

In the next installment, we’ll discuss many more fun facts about cyclotomic fields.


[1] Goldstein, Larry Joel – Analytic Number Theory

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