Is the Galois group Gal$(\mathbb{Q}(2^{\frac{1}{8}}, i)/\mathbb{Q}(i))$ cyclic?

Question

Is the Galois group Gal$(\mathbb{Q}(2^{\frac{1}{8}}, i)/\mathbb{Q}(i))$ cyclic?

I started by the fact that $x^8-2$ is the minimal polynomial. But then the $4th$ roots of unity are involved, so I am not sure how the extension is even galois. I must be making a mistake. Also, even if I could show that it is indeed a Galois extension, what's the best way to show that it is cyclic or not?