Cyclotomic polynomials

Cyclotomic (or "circle dividing") polynomials are the irreducible factors of $z^{n}-1\!$ when the polynomial coefficients are restricted to the field of rational numbers. For example

$z^{6}-1=(z-1)(z+1)(z^{2}+z+1)(z^{2}-z+1)$.

The cyclotomic polynomials are often written as $\Phi _{n}(z)$. The polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^n - 1\!} has exactly as many of these factors as there are integer divisors of n:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^6 - 1 = \Phi_1(z)\Phi_2(z)\Phi_3(z)\Phi_6(z) } .

Finding cyclotomic polynomials

Recursive method

Cyclotomic polynomials are defined by the equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^n - 1 = \prod_{d\,\mid\,n} \Phi_d(z)} .

Solving for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_n(z) } , one obtains

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_n(z) = \frac {z^n - 1} {\prod_{(d \ne n)\,\mid\,n} \Phi_d(z)} } .

One can then construct any polynomial recursively, given that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_1(z) = z - 1} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_2(z) = \frac {z^2 - 1} {z-1} = z + 1 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_3(z) = \frac {z^3 - 1} {z-1} = z^2+z+1 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_6(z) = \frac {z^6 - 1} {(z-1) (z+1) (z^2 + z + 1)} = z^2-z+1 } .

Direct method

For larger values of n, it is easier to use the direct formula, based on the Möbius inversion of the above formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_n(z)=\prod_{d\,\mid n}(z^{n/d}-1)^{\mu(d)},}

where μ is the Möbius function. For example, one would follow these steps for n = 60:

1. Factor n.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 60 = 2^2*3*5 }

2. Determine the square-free divisors of n, and classify them according to whether their number of prime factors are even or odd.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 30 = 2*3*5 } (Odd)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 15 = 3*5 } (Even)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10 = 2*5 } (Even)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6 = 2*3 } (Even)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5= 5 } (Odd)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3= 3 } (Odd)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2= 2 } (Odd)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1= 1 } (Even, since 1 is not a prime)

3. Form the factors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^{n/d} - 1 } , putting the factors from the even class in the numerator and those from the odd class in the denominator.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{60}(z) = \frac { (z^{60}-1) (z^{10}-1) (z^6-1) (z^4-1) } { (z^{30}-1) (z^{20}-1) (z^{12}-1) (z^2-1) } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac { (z^{30}+1) (z^2+1) } { (z^{10}+1) (z^6+1) } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac { z^{20}-z^{10}+1 } { z^4-z^2+1 } }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = z^{16}+z^{14}-z^{10}-z^8-z^6+z^2+1 } .

Construction from complex roots

It is also possible to construct a cyclotomic polynomial from the complex roots of 1:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_n(z) = \prod_{k=1}^{\varphi(n)}(z-e^{\frac {2\pi k} {n} j})\;}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(n)} is Euler's totient function. For example,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_6(z) = (z-e^{\frac {\pi} {3} j})(z-e^{\frac {5\pi} {3} j}) = z^2-z+1.}

Some special cases

When p and q are odd primes,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_p(z)=\frac{z^p-1}{z-1}=\sum_{k=0}^{p-1} z^k}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{2p}(z)=\frac{z^p+1}{z+1}=\sum_{k=0}^{p-1} (-z)^k}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{pq}(z)=\frac{(z^{pq}-1) (z-1)}{(z^p-1) (z^q-1)}=\frac {\sum_{k=0}^{q-1} z^{pk} } {\sum_{k=0}^{q-1} z^k}}

Galois Theory

The Galois group associated to the nth cyclotomic polynomial is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}/n\mathbb{Z})^*} , the group of cyclic units modulo n. Cyclotomic fields play an important part in algebraic number theory and can be used to show that certain constructions posited by ancient Greek geometers are impossible, such as trisecting the angle and constructing certain regular polygons.