n-TH root selections in fields

Abstract

In this work we generalize the results of [9] to the higher level case: we define n-th root selections in fields of characteristic 6= 2, that is subgroups of the multiplicative group of a field whose existence is equivalent to the existence of a partial inverse of the x 7! xn function, provide necessary and sufficient conditions for such a subgroup to exist, study their existence under field extensions, and give some structural results describing the behaviour of maximal n-th root selection fields.