Mathematical Meanderings: Cyclotomic Polynomials

Recently I turned 26142 days and this number has the property that it is a member of OEIS A138938:

Indices k such that A019326(k)=$ \Phi_8$ is prime, where $ \Phi $ is a cyclotomic polynomial.

I've heard many times about cyclotomic polynomials over the past five years without really understanding their significance. This occasion provided an opportunity to investigate the topic further. I discovered that SageMath (which I use for most of my calculations) has a function to generate the cyclotomic polynomials. The following simple command will generate the first twenty cyclotomic polynomials:

for n in [1..20]:
print(n,"-->",cyclotomic_polynomial(n,x))

$ \Phi_1 = x - 1$

$ \Phi_2 = (x + 1$

$ \Phi_3 = x^2 + x + 1$

$ \Phi_4 = x^2 + 1$

$ \Phi_5 = x^4 + x^3 + x^2 + x + 1$

$ \Phi_6 = x^2 - x + 1$

$ \Phi_7 = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$

$ \Phi_8 = x^4 + 1$

$ \Phi_9 = x^6 + x^3 + 1$

$ \Phi_{10} = x^4 - x^3 + x^2 - x + 1$

$ \Phi_{11} = x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$

$ \Phi_{12} = x^4 - x^2 + 1$

$ \Phi_{13} = x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$

$ \Phi_{14} = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1$

$ \Phi_{15} = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1$

$ \Phi_{16} = x^8 + 1$

$ \Phi_{17} = x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$

$ \Phi_{18} = x^6 - x^3 + 1$

$ \Phi_{19} = x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$

$ \Phi_{20} = x^8 - x^6 + x^4 - x^2 + 1$

But what are the cyclotomic polynomials, or cyclotomic polynumbers as they are also called, and why are they important? A definition of a cyclotomic polynumber is that it is an irreducible polynumber with integer coefficients, $n=1,2,3,...$ such that:$$ 1-x^n=\prod_{d|n} \Phi_d \text{ where } d\text{ are the divisors of } n$$By irreducible is meant they cannot be factored into a product of polynomials of lesser degree. In this respect, these polynumbers are like prime numbers. The name cyclotomic arrives from the view of these polynumbers being solutions to $1-z^n$ on the complex plane. The solutions lie on the unit circle and can be thought of as cutting up this circle into $n$ sectors (see Figure 1 that shows the solutions to $1-z^{10}=1$).

Figure 1

From the earlier definition it can be seen that:$$1-z^{10}=\Phi_1 \times \Phi_2 \times \Phi_5 \times \Phi_{10} \text{ and so}$$ $$1-z^{10}=(z-1)(z+1)(z^4 + z^3 + z^2 + z + 1)(z^4 - z^3 + z^2 - z + 1)$$There are some other interesting results, including:$$ \Phi_p=1+x+x^2+...+ \,x^{p-1} \text{ where } p \text{ is prime }$$Looking at the cyclotomic polynumbers shown at the beginning of this post, it can be seen that $ \Phi_2, \Phi_3, \Phi_5, \Phi_7, \Phi_{11}, \Phi_{13}, \Phi_{17} \text{ and } \Phi_{19} $ follow this pattern. Another interesting result is that:$$ \Phi_{p^k}=\Phi_p(x^{p^{k-1}}) \text{ with } p \text{ again prime }$$An example illustrating the previous result is: $$\Phi_9=\Phi_{3^2}=\Phi_3(x^3)=1+x^3+(x^3)^2 + 1=1+x^3+x^6$$Another property of cyclotomic polynumbers is that:$$ \Phi_{pm} \Phi_m=\Phi_m(x^p) \text{ where } p \text{ is prime and gcd}(p,m)=1$$This result could be used to find $ \Phi_{60}$ in terms of lesser polynumbers because 60 = 5 x 12 and so:$$\Phi_{60}\; \Phi_{12}=\Phi_{12}(x^5) = (x^5)^4 - (x^5)^2 + 1 = x^{20}-x^{10}+1$$ $$ \Phi_{60}= \frac{x^{20}-x^{10}+1}{x^4 - x^2 + 1}=1+x^2-x^6-x^8-x^{10}+x^{14}+x^{16}$$Another useful result arising from the previous result as the case where $p=2$ is:$$ \Phi_{2m} \; \Phi_m=\Phi_m(x^2) \text{ where } m \text{ is odd }$$ $$ \text{ Hence } \Phi_{2m}= \frac{\Phi_m(x^2)}{\Phi_m}$$As a particular example, let's work out $ \Phi_{14} $ where of course 14 = 2 x 7:$$ \Phi_{14}=\frac{\Phi_7(x^2)}{\Phi_7}=\frac{1+x^2+x^4...+ \, x^{12}}{1+x+x^2+...+ \, x^6}=1-x+x^2-x^3+x^4-x^5+x^6$$Lastly, for this post at least, it can be noted that there is a connection with Euler's totient function via the relationship:$$ \text{ degree of } \Phi_n=\phi(n) \text{ where } \phi \text{ is Euler's totient function}$$We can see this result at work if we look at say $ \Phi_{12} = x^4 - x^2 + 1$ which has degree 4 and $\phi(12)=4 $ since the numbers that are coprime to 12 are 1, 5, 7 and 11. I'd like to thank N. J. Wildberger's Insights into Mathematics YouTube channel for providing some of the examples and information in this blog post. Here is a link to this channel's videos on Cyclotomic Polynomials:

Cyclotomic polynumbers fall into the category of algebraic number theory and what's interesting about N. J. Wildberger's approach is that he doesn't feel complex numbers (falling into the category of Complex Analysis) should be used to explain their properties. Of course, I've only scratched the surface of this topic but at least I've made a start that I can build on in later posts.

Mathematical Meanderings

Monday, 2 November 2020

Cyclotomic Polynomials

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