Algebraic function fields with small class number

Classification of Algebraic Function Fields with Divisor Class Number Two

“…Previously MacRae [5] gave the quadratic solutions which have a place of degree one. The divisor class number two problem for algebraic function fields has been studied previously in [4], but we do not agree with some results contained in that paper. In [3] we proved that up to isomorphism there are exactly 11 quadratic algebraic function fields such that h ϭ 2.…”

“…The first two function fields are such that h ϭ 1 and they were given in [4]. From this classification we know that there are only 3 solutions to our problem.…”

“…We recall that h ϭ L (1), where L is the numerator polynomial of the zeta function of K/k. The classification of function fields K/k such that h ϭ 1 is done in [6] and [4]. Previously MacRae [5] gave the quadratic solutions which have a place of degree one.…”

Five wild nearfields

“…We claim that for every vertex or edge x of Y , there exists a unique epimor- [8]. This ring R has 3 units, hence PSL 2 R is not a Frobenius group on the corresponding projective line.…”

On real quadratic function fields of Chowla type with ideal class number one

“…Madan, C.S.Queen and R.E.MacRae [5,6,7] determined all imaginary quadratic function fields K (even for the case 2|q) with h(O K ) = 1. For the real case, we can ask the following question as an analogy of a Gauss conjecture: Are there infinitely many of real quadratic function fields K with h(O K ) = 1 for any fixed q?…”