Monday, 12 June 2017

galois theory - If a cyclotomic polynomial is reducible over a finite field, what does its factorisation look like?



The nth cyclotomic polynomial remains irreducible when reduced modulo p if and only if p is a generator of Z×n. Suppose that is not the case, and I know that the polynomial can be factored over Fp. What can I say about the degrees of the irreducible factors?



For example, the 13th cyclotomic polynomial is reducible modulo 3, since 331 modulo 13. A (long, tedious) factorisation attempt reveals that there are four cubic irreducible factors. Should I have known this a priori?


Answer



Let p. Then the cyclotomic polynomial \Phi_n factors over \Bbb F_p
into \phi(n)/r irreducible factors each of degree r, where r

is the multiplicative order of p modulo n.



To see this, consider a
primitive n-th root of unity \zeta in an extension
of \Bbb F_p. The number of conjugates of \zeta over \Bbb F_p is the least positive integer r with \zeta^{p^r}=\zeta, that is p^r\equiv1\pmod n.


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