Computing cyclotomic polynumbers
Insights into Mathematics
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The cyclotomic polynomials -- or polynumbers as we prefer to call them --- (called Phi_n) are an important family arising from a simple factorization problem, but having major applications to algebra, number theory and indeed also to geometry. Initially they arise from the problem of factoring the polynomial, or polynumber 1-xn for some natural number n.
In some sense they are also rather mysterious, as it is not at all easy to calculate Phi_n if n is a large natural number. They are also special because they are irreducible, and it is not so easy to quickly generate irreducible polynomials of high degree.
We will motivate the discussion by looking for Mersenne primes, and get to a statement of the Schoenemann- Eisenstein criterion for irreducibility.