On certain character sums over $\mathbb F_q[T]$

Let ${\mathbb F}_{q}$ be the finite field with $q$ elements and let $\mathbf{A}$ denote the ring of polynomials in one variable with coefficients in ${\mathbb F}_{q}$. Let $P$ be a monic polynomial irreducible in $\mathbf{A}$. We obtain a bound for the least degree of a monic polynomial irreducible in $\mathbf{A}$ ($q$ odd) which is a quadratic non-residue modulo $P$. We also find a bound for the least degree of a monic polynomial irreducible in $\mathbf{A}$ which is a primitive root modulo $P$.