examples of cyclotomic polynomials
In this entry we calculate a number of cyclotomic polynomials, , for various . The interested reader can find specific examples at the bottom of the entry. We will concentrate first on the theory details which allow us to calculate these polynomials.
1 The theory behind the examples
The following simple lemma is also useful when calculating cyclotomic polynomials:
Let be positive integers. If divides then divides .
and so, is also an th root of unity and, in particular, it is a root of . By the properties of minimal polynomials, must divide . ∎
The polynomial is of degree , where is Euler’s phi function.
We begin with the th cyclotomic polynomials for a prime .
is irreducible over .
In order to show that is irreducible, we perform a change of variables , and define . Clearly, is irreducible over if and only if is irreducible. Also:
Since all the binomial coefficients , for , are integers divisible by , and is not divisible by , we can use Eisenstein’s criterion to conclude that is irreducible over . Thus is irreducible as well, as desired. ∎
As a corollary, we obtain:
Let be a prime. Then the th cyclotomic polynomial is given by
By the lemma, the polynomial divides and, by the proposition above, is irreducible. Hence as claimed. ∎
The following proposition will be very useful as well:
A proof can be found in this entry (http://planetmath.org/FactorsOfNAndXn1). ∎
2 The examples
A generous list of examples can be found in this entry (http://planetmath.org/PrimeFactorsOfXn1). The examples of can be calculated by recursively factoring the polynomials , for , using (a) the fact that for primes and (b) the polynomial is a divisor of if and only if is a multiple of (and appears with multiplicity one as a factor, because does not have repeated roots). Hence, we can calculate:
Before factoring , note that we know that divides it, divides it and has as many divisors as . Therefore .
The polynomial is (by the Theorem). In order to calculate we factor . Once again, note that has positive divisors, and we already know the following divisors: , , . Hence:
Notice that we knew a priori (by a Lemma above) that the degree of is in fact . Similarly, suppose we want to calculate . This is a polynomial of degree , and divides . On the other hand, has irreducible factors and we already know the factors corresponding to . Thus:
Incidentally, we can find an explicit root of in terms of radicals. The roots are simply given by:
|Title||examples of cyclotomic polynomials|
|Date of creation||2013-03-22 17:20:03|
|Last modified on||2013-03-22 17:20:03|
|Last modified by||alozano (2414)|
|Synonym||calculating cyclotomic polynomials|