Genus fields of congruence function fields

Abstract

ABSTRACT. Let k be a rational congruence function field and consider an arbitrary finite separable extension K/k. If for each prime in k ramified in K we have that at least one ramification index is not divided by the characteristic of K, we find the genus field Kge, except for constants, of the extension K/k. In general, we describe the genus field of a global function field.

Genus fields of Kummer extensions of rational function fields

“…From Theorem 3.6, we know each (K j ) ge , 1 ≤ j ≤ h. The knowledge of K ge would follow if we had K ge = h j=1 (K j ) ge . However in general for any two fields L 1 and L 2 , we only have that (L 1 ) ge (L 2 ) ge ⊆ (L 1 L 2 ) ge (see [6] for an example where (L 1 ) ge (L 2 ) ge (L 1 L 2 ) ge ). Now, in our case we have…”

Class fields, Dirichlet characters, and extended genus fields of global function fields

“…In fact, they defined the extended genus field for an arbitrary global function field 𝐾 using a generalization of cyclotomic function field extensions given by the Carlitz module. The genus field of a general abelian extension of a global function field over a rational function field using Dirichlet characters can be found in [2,15,16].…”

“…This is Theorem 4.5, where we obtain that 𝐸 gex is the field associated with a group of Dirichlet characters. In particular, this definition is the same as the one given in [18] in the case of a field contained in a cyclotomic function field but it might be different in the general case, even for abelian extensions.…”

Function field genus theory for non-Kummer extensions

“…In [3], H. Leopoldt studied the extended genus field K gex of a finite abelian extension of the field of rational numbers Q, by means of Dirichlet characters. Using Leopoldt's technique we applied Dirichlet characters to the function field case and found a general description of K ge ( [1,4,5]). In these papers it was also provided an explicit description of K ge in the cases of a Kummer cyclic extension of prime degree l and of an abelian p-extension where p is the characteristic of k. In [2,6,8], the explicit description of K ge was given when K/k is a finite Kummer l-extension with l a prime number.…”