The exponent of class groups on congruence function fields

An explicit triangular integral basis for any separable cubic extension of a function field

“…We note that this work generalises the results begun in [6,Theorem 5.17], and that our work in characteristic p ě 5 is a consequence of our computations and results appearing in [4,8]. In characteristic 3, the basis we construct bears some resemblance to that of [5,Theorem 9], although the construction is different.…”

Explicit genus formula for any separable cubic global function field

“…Thus, as v p pa 1 q ă 0, we obtain v p pja 2 1 `pw 3 2 `a1 w 2 qq ě mintv p pja 2 1 `w3 2 q, v p pa 1 w 2 qu ą mintv p pja 2 1 q, v p pa 1 w 2 qu " mintv p pja 2 1 q, v p pa 1 q `2v p pa 1 q{3u " mint2v p pa 1 q, 5v p pa 1 q{3u " 2v p pa 1 q. Hence v p pa 2 q " v p ˆpja 2 1 `pw 3 `a1 wqq 2 a 3 1 ˙ą 4v p pa 1 q ´vp pa 3 1 q " v p pa 1 q.…”

“…Then v p pα ´3ja 2 1 q " 0. Let w 0 P K be chosen so that w 0 ‰ ´α´3 ja 2 1 and v p pα ´3ja 2 1 `w0 q ą 0. This may be done via the following simple argument: As v p pα ´3ja 2 1 q " 0, then α ´3ja 2 1 ‰ 0 in kppq.…”

Congruence function fields of genus 𝑔 and class number 𝑔+1

“…In his recent doctoral thesis, D. J. Madden [5] has proved that if a is a primitive integral element in a geometric extension F of K(x) of prime power degree /", where / + char K, and containing /" -1 roots of unity there is a prime px lying over the infinite prime of K(x) so that…”