Math 514 homework #17 Due date: 19951030 D. J. Bernstein 19951023 Let f = f_{d}x^{d} + f_{d−1}x^{d−1} + ⋯ + f_{0} and g = g_{e}x^{e} + g_{e−1}x^{e−1} + ⋯ + g_{0} be polynomials over some ring. The following matrix, with d + e rows and d + e columns, is Sylvester’s matrix for f and g:

( \begin{matrix}f_{d} & f_{d−1} & f_{d−2} & ⋯ & f_{1} & f_{0} & 0 & 0 & ⋯ & 0 \\ 0 & f_{d} & f_{d−1} & f_{d−2} & ⋯ & f_{1} & f_{0} & 0 & ⋯ & 0 \\ 0 & 0 & f_{d} & f_{d−1} & f_{d−2} & ⋯ & f_{1} & f_{0} & ⋯ & 0 \\ ⋮ & ... & ... & ... & ... & ... & ... & ... & ... & ⋮ \\ 0 & 0 & 0 & 0 & f_{d} & f_{d−1} & f_{d−2} & ⋯ & f_{1} & f_{0} \\ g_{e} & g_{e−1} & g_{e−2} & ⋯ & g_{0} & 0 & 0 & 0 & ⋯ & 0 \\ 0 & g_{e} & g_{e−1} & g_{e−2} & ⋯ & g_{0} & 0 & 0 & ⋯ & 0 \\ 0 & 0 & g_{e} & g_{e−1} & g_{e−2} & ⋯ & g_{0} & 0 & ⋯ & 0 \\ 0 & 0 & 0 & g_{e} & g_{e−1} & g_{e−2} & ⋯ & g_{0} & ⋯ & 0 \\ ⋮ & ... & ... & ... & ... & ... & ... & ... & ... & ⋮ \\ 0 & 0 & 0 & 0 & 0 & g_{e} & g_{e−1} & g_{e−2} & ⋯ & g_{0} \end{matrix} )

Here the coefficients of f are repeated on e rows, and the coefficients of g are repeated on d rows. The determinant of Sylvester’s matrix is the resultant of f and g, written result(f, g). Exercise 52. Let f and g be polynomials over some ring R. Prove that result(f, g) ∈ fR[x] + gR[x]. (Hint: Apply Sylvester’s matrix to the transpose of (x^{d+e−1}, x^{d+e−2}, …, x, 1).) Let m be a positive integer. A complex number ζ satisfying ζ^{m} = 1 is called an mth root of unity. If ζ does not satisfy ζ^{k} = 1 for any positive integer k < m, ζ is called a primitive mth root of unity. In all subsequent exercises, ζ_{m} means exp(2πi/m). Thus ζ_{m}^{m} = 1. The powers ζ_{m}^{a} for a ∈ {0, 1, …, m − 1} are distinct. Exercise 53. Prove that x^{m} − 1 = ∏_{0≤a≤m−1}(x − ζ_{m}^{a}). (Hence the powers ζ_{m}^{a}, for a ∈ {0, 1, …, m − 1}, are the only mth roots of unity.) Exercise 54. Prove that ζ_{m}^{a} is a primitive dth root of unity if and only if gcd {a, m} = m/d. (In particular, ζ_{m}^{a} is a primitive mth root of unity if and only if a and m are coprime.)