I am stuck here please should be detail work!

Q1

Let F, be a finite field with q = p" elements, where p is a prime. Let II : F. - F, be the Frobenius automorphism II(x) = 2. Prove that II considered as a linear map over F, is diagonalizable if and only if n divides p" - 1. (Here is a misprint. It should be "n divides p - 1". )Let A(t) be a non-singular matrix whose elements are differentiable func- tions of real variable t. Let A'(t) denote the matrix formed by the derivatives of the elements. Show that the derivative of the determinant det A satisfies dt (det A) = det A . trace( A' . A-1).Let V be the vector space of polynomials p(x) = a + br + cr with real coefficients a, b, and c. Define an inner product on V by (P. q) = p(x)q(z)dr. (a) Find an orthonormal basis for V consisting of polynomials do( I), $1(I), and $2(x), having degree 0, 1, and 2, respectively. (b) Use the answer to (a) to find the second degree polynomial that solves the minimization problem min (p(z) - 13)'dr. PEV -1