Rationalizing fractions over finite fields

Homework Statement

Let w be a primitive n-th root of unity in some finite field. Let 0 < k < n. My question is how to rationalize

[\tex]\dfrac{1}{1 + w^k}[\tex].

That is, can we get rid of the denominator somehow? I know what to do in the case of complex numbers but here I'm at a loss. Thanks!