finite fields question

[tarik]

Let k be a positive divisor of q-1 and a be an element of the finite field F_q. Prove that the equation

x^k = a has a solution in F_(q^k), where F_(q^k) is a finite extension of F_q.

Since k divides q-1, we have q = k*m + 1 where m is a positive integer. Also a is in F_q, thus a^q = a in F_q. That is, a^(k*m + 1) = a in F_q. Now how can we proceed on the extension field F_(q^k) ? Should we use Hilbert's 90's Thm? If yes, how? Please help!