Why is $K((X))$ free module over $K((X^q))$ with basis $1,X,X^2,・・・,X^{q-1}$?

Question

Let $K$ be a field. Why is $K((X))$ free module over $K((X^q))$ with basis $1,X,X^2,・・・,X^{q-1}$ ?

I want to prove this.

My try : [$K((X)):K((X^q))]＝q$, so $1,X,X^2,・・・,X^{q}$ is linearly dependent.