### Computing the Galois action of a generator on an unit of a subfield

Hi! The set up for my question is the following:

Let,

- k be a quadratic number field,
- K|k be a unramified cyclic extension of degree 5 over k,
- L a subfield of absolute degree 5 of K,
- G the Galois group of K, which in this case is the dihedral of order 10

If I know k and L, is there is a way to compute the action of a generator of G on a unit of L without having to compute K and then G? If it's not possible, what would be the correct way to do this?