A field has a unique algebraic extension of each degree if and only if its absolute Galois group is the profinite completion of integers

Question

I would like to know how to prove that a field $K$ which has a unique algebraic extension of each degree has its absolute Galois group isomorphic to the profinite completion of the integers $\hat{\mathbb{Z}}$.

One can probably reduce the problem to showing that the Galois groups of finite algebraic extensions of $K$ of each degree is a cyclic group, but I do not know how to proceed.