Double coverings for quadratic extensions and function fields

A double covering of a Galois extension K/k in the sense of P. Das (2000) [4], is an extension of degree ⩽2 such that is Galois. In this paper we determine explicitly all double coverings of any quadratic extensions, the Carlitz cyclotomic extensions of the rational function field over a finite field, and their maximal real subfields. In the case of quadratic extensions, we get the result by Hilbert Theorem 90 and in the function field cases, we get the results by using the method in G.W. Anderson (2002) [1], with a modification and using the results in S. Bae et al. (2003) [2], and S. Bae and L. Yin (2004) [3]. We also construct explicitly a large kind of (q−1)-th coverings of cyclotomic function fields.