2015/12/16 by Nguyen, Dong Quan Ngoc
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1512.05043
Hilbert's Satz 90 tells us that for a given cyclic extension K/k, a unit of norm 1 in K can be written as a quotient of conjugate elements in K. For the extensions ℚ(ζp)/ℚ with p prime > 3, Newman proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary condition for which a unit of norm 1 in ℚ(ζp) can be written as a quotient of conjugate units. In order to obtain this result, Newman proved a stronger result that gives a unique representation of units of norm 1 as a product of a power of \dfrac1 - ζpe1 - ζp with a quotient of conjugate units, where e is a given primitive root modulo p. In this paper, we obtain a function field analogue of Newman's result for the \wp-th cyclotomic function field extensions \mathbbK\wp/\mathbbFq(T), where \wp is a monic prime in \mathbbFq[T]. As a consequence, we proved a refinement of Hilbert's Satz 90 for the extensions \mathbbK\wp/\mathbbFq(T) that gives a sufficient and necessary condition for which a unit of norm 1 in \mathbbK\wp can be written as a quotient of conjugate units.