Re: [math-fun] Expansion of charpoly's in terms of traces ?
<<
I'm not aware of mathematicians who define polynomials over finite fields with respect to Fermat's Little Theorem.
Indeed they don't --- that's exactly the problem!
The polynomial ring over any field F is denoted by F[X]. It is a mathematical object that one may or may not have any interest in. If one is more interested in what becomes of F[X] after X^p is identified with X, then one need only consider the quotient ring F[X] / (X^p - X), where (X^p - X) denotes the ideal generated by X^p - X. I don't see any problem. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On 12/4/09, Dan Asimov <dasimov@earthlink.net> wrote:
<<
I'm not aware of mathematicians who define polynomials over finite fields with respect to Fermat's Little Theorem.
Indeed they don't --- that's exactly the problem!
The polynomial ring over any field F is denoted by F[X]. It is a mathematical object that one may or may not have any interest in.
If one is more interested in what becomes of F[X] after X^p is
identified with X, then one need only consider the quotient ring
F[X] / (X^p - X),
where (X^p - X) denotes the ideal generated by X^p - X.
I don't see any problem.
My point is precisely that one should _not_ be interested in "what becomes of F[X] after X^p is identified with X", because the set of functions of integers which can be represented by members of the quotient ring is artificially restricted by its kernel! In contrast, the binomial function basis permits "polynomials" of all degrees. WFL
participants (2)
-
Dan Asimov -
Fred lunnon