Properties of Conjugate Reciprocal Polynomials
If p(z) is the minimal polynomial of z0 with |z0| = 1, and p(z) has real coefficients, then p(z) is self-reciprocal. This follows because
- .
So z0 is a root of the polynomial which has degree n. But, the minimal polynomial is unique, hence
for some constant c, i.e. . Sum from i = 0 to n and note that 1 is not a root of p. We conclude that c = 1.
A consequence is that the cyclotomic polynomials are self-reciprocal for ; this is used in the special number field sieve to allow numbers of the form, and to be factored taking advantage of the algebraic factors by using polynomials of degree 5, 6, 4 and 6 respectively - note that (Euler's totient function) of the exponents are 10, 12, 8 and 12.
Read more about this topic: Reciprocal Polynomial
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—Ralph Waldo Emerson (1803–1882)
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