Regular Normal Extensions over Complete Fields

A general Kummer theory for function fields

“…If D becomes principal in a normal extension K, it already becomes principal in some radical extension K C K. If there is some e-th root of unity not present in F, this root appears at least once as the value of a character C(r) for Kr. The equations (14) and (I) then show that the Galois group of K' is not Abelian. Hence our condition is necessary.…”

“…Such an extension may be described by choosing in G for each automorphism a of F a representative U which induces a. Every element of G then has the form SU, for some S e H, e F. If H is Abelian, the multiplication of these elements is determined by a table (14) UTU-T (T H, ( I'), (15) UU S.U (each S. e H).…”

“…Here (14) indicates that transformation by each a determines an automorphism T --T of H; it is independent of the choice of U and has (T) T. In (15) S, is a factor set 1 of r in H. We wish to find the special conditions holding for these automorphisms and factor sets in the case when K is a radical extension. THEOREM 14.…”

“…If the Abelian group H is finite, and has a basis of auto- One may also observe that the conditions (I) and (II) do not depend essentially upon the choice of the representatives U used in (14) and (15) to describe the group G; the validity of (I) and (II) for any one set of such representatives implies the validity for any other set.…”

“…THEOREM 15. Let a finite group G be represented as in (14) and (15) as an extension of its normal subgroup H by the factor group F G/H, where (i) F is the Galois group of a finite separable normal extension of , (ii) H is an Abelian group with exponent e prime to the characteristic of , (iii) contains all e-th roots of unity, (iv) The group X of characters of H is isomorphic to a group D/(F) of divisor classes from F. Then G can be realized as the Galois group of a normal separable radical extension K/F with coecient field if and only if each character C of H in satisfies the conditions (I) and (II) of Theorem 14. When these conditions hold, a field K may be so constructed that D is the group of all divisors which become principal in K.…”