Bill Daly on Fri, 23 Jan 2009 07:10:54 +0100

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Polmod factorization

If f(x) is an irreducible polynomial in x, then Mod(x,f(x)) is a generic root of f(x), and the algebra mod f(x) is isomorphic (I think) to the algebra of the field generated by appending any root of f(x) to Q. Is there a way of factoring f(x) mod f(x)? What I have in mind is that for some polynomials where Mod(x,f(x)) is a root, then there may be other rational functions of x which are also roots of f(x), e.g. if f(x) is polcyclo(n), then Mod(x^a,f(x)) is a root whenever a is coprime to n. I don't however see any easy way of finding such roots with polmods in PARI. What, if anything, am I overlooking?

Regards, Bill