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My students found an exercise about cyclotomic polynomials in the AOPS precalculus text. They asked me why this construction exists in the first place and what it's good for... I am looking to give them a more satisfying answer than: "We use it in the study of Galois theory." They are familiar with the symmetry group $S_3$ and are very talented $7/8$th graders.

What's a good answer to give them?

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    $\begingroup$ Something like "First, this is one of the few broad families of polynomials we can really substantially understand. Second, they arise in practice in many different circumstances"? Probably those reasons are unfamiliar and unsatisfying to very bright young kids, though. $\endgroup$ – paul garrett Dec 21 '18 at 22:30
  • $\begingroup$ The first point you make is strong. I can point out that this construction is understandable in a way that many other polynomials may not be and point out how difficult polynomials (and finding their zeros) can be in less manageable constructions. If I say "many different circumstances..." I will get the response you might expect from a young person... "Many circumstances... like what?" $\endgroup$ – Mason Dec 21 '18 at 23:16
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    $\begingroup$ These are 7th/8th grade students (so: ages ~12-14) who are doing AoPS precalc, reading about cyclotomic polynomials, and are familiar with $S_3$? I would think such students have quite a bit of intrinsic motivation as far as exploring mathematical content. Perhaps you could ask them to look into the history/use of cyclotomic polynomials and tell you why this is an object/concept worth studying. $\endgroup$ – Benjamin Dickman Dec 21 '18 at 23:49
  • $\begingroup$ Also, a "fun fact" about roots of unity is that they are expressible in radicals... I always loved that point... (cf. Lagrange resolvents...) $\endgroup$ – paul garrett Dec 21 '18 at 23:52
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    $\begingroup$ The comment by @paulgarrett can be combined with the (possibly fun) fact that solvability by iterated square roots amounts to ruler-and-compass constructibility. You can have fun with the golden ratio and its connections with regular pentagons. Then you can tell them about 17. $\endgroup$ – Andreas Blass Dec 22 '18 at 0:44
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Can we grant that the students think roots of unity are worthwhile? If so, point out that one of the ways to understand roots of polynomials (like roots of $x^n-1$) is to understand the lowest degree polynomial with those roots. The $n$ roots of unity do not all behave in the same way algebraically, but the primitive $n$th roots of unity (in $\mathbf C$) do. That is what irreducibility of the cyclotomic polynomials tells us.

Here are two applications of these polynomials.

  1. They have a role in an elementary proof that for each $n>1$ there are infinitely many primes $p$ such that $p\equiv 1 \bmod n$ (Euclid's proof that there are infinitely many primes uses the 2nd cyclotomic polynomial.)

  2. They have a role in Witt's proof of Wedderburn's theorem that all finite division rings are commutative.

The cyclotomic polynomials are a nice example of a pattern that lasts a long time but eventually stops: the first 100 of them have coefficients in {0,1,-1}, but farther out this is not true (first instance is the 105th cyclotomic polynomial).

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On the topic of Sicherman dice: "Cyclotomic polynomials play an important role in analyzing dice labelings of six-sided dice as well as other sizes of dice."

http://buzzard.ups.edu/courses/2010spring/projects/jenkins-sicherman-dice-ups-434-2010.pdf

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