Degree $8$ cyclic extension over $\mathbb{Q}$

Question

Actually I am interested in degree $ 8 $ cyclic extension over $ \mathbb{Q} $. Let $ L $ be such extension. At first I was thinking to take basis as normal basis, as we can determine the galois group and how the group acted on $ L $ but we don't know how the multiplication can be defined. Next to make easy calculation in multiplication I take primitive element as this is simple extension but we dont know how the Galois group will act on $ L $. Then I consider a particular cyclic extension, $ \mathbb Q(\eta) $ where $ \eta $ be $ 17^{\text{th}} $ primitive roots of unity. Then by galois fundamental theorem we have an extension(cyclic Galois) $ L $ over $ \mathbb{Q} $ of degree $ 8 $. Now in this extension we can determine the multiplication rule and the action of galois group also, but multiplication between two elements like $b $ and $\sigma(b)$ are so much calculative. So does there any cyclic extension $ L $ over $ \mathbb{Q} $ where we know the multiplication rule and also the galois group action and also the calculation is easier. I.e we can calculate the term $ b\sigma(b) $ easily where $ b \in L $.