Exercise 8, § 12

Algebra, V, p. 168

Let K be a finite field of q elements and

P(X) = a_0 + a_1 X + \cdots + a_{q-2} X^{q-2}

a polynomial of $K[X]$ of degree $q-2$ . Let r be the rank of the matrix

A = \begin{pmatrix} a_0 & a_1 & \cdots & a_{q-2} \\ a_1 & a_2 & \cdots & a_0 \\ \ldots & \ldots & \ldots & \ldots \\ a_{q-2} & a_0 & \cdots & a_{q-3} \end{pmatrix}.

Show that the number of distinct roots of P in K, other than 0, is $q-1-r$ . (Consider the product VA, where V is the Vandermonde matrix of elements $x_1, x_2, \ldots, x_{q-1}$ of $K^*$ .)

Solution

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