Abstract. Let Fq be the finite field with q elements, (2 |q),with deg A(x) ≥ 1 and a ∈ F * q . In this paper several equivalent conditions for the ideal class number h(O K ) to be one are presented and all such quadratic function fields with h(O K ) = 1 are determined.

In [Algebra and Discrete Mathematics, 2 (2003), [32][33][34][35], Kehayopulu N., Ponizovskii J. and Tsingelis M. showed that in commutative semigroups (resp. ordered semigroups) with identity, every maximal ideal is a prime ideal and the converse is not true in general. In this note we prove that for an ordered semigroup S satisfying S = (S 2 ] all maximal ideals are weakly prime ideals. This generalize the corresponding result in the above paper. We prove that S = (S 2 ] is necessary. Some conditions under which prime ideals are maximal are given.It is well known that in commutative rings having an identity, every maximal ideal is a prime ideal, but the converse does not hold. So it is interesting whether it is still valid for semigroups and for ordered semigroups as well. In [1], it is showed that in commutative semigroups (resp. ordered semigroups) with identity, each maximal ideal is a prime ideal and the converse does not hold in general.In this note we prove that for an ordered semigroup S satisfying S = (S 2 ] all maximal ideals are weakly prime. Since in commutative ordered semigroups all weakly prime ideals and prime ideals are the same [1], in commutative ordered semigroups S satisfying S = (S 2 ] all maximal ideals are prime ideals. This generalize the corresponding result in [2]. We prove that S = (S 2 ] is necessary. In the paper we also give some conditions under which all prime ideals are maximal.Recall that a semigroup S that is also a partially ordered set, in which the binary operation and the order relation are compatible, is called an ordered semigroup. A non-empty subset I of ordered semigroup S is called an ordered ideal of S if (i) SI ⊆ I,Maximal ideals of ordered semigroup have been studied in [3][4][5][6].In this paper, the notion A = {I j |j ∈ J} denotes the set of all proper ideals of ordered semigroup S, Q = {Q α |α ∈ Γ} denotes the set of all proper prime ideals of S, M = {M α |α ∈ Λ} denotes the set of all maximal ideals of S, A * = ∩{I j |j ∈ J} denotes the intersection of all proper ideals of S, Q * = ∩{Q α |α ∈ Γ} denotes the intersection of all proper prime ideals of S, and M * = ∩{M α |α ∈ Λ} denotes the intersection of all maximal ideals of S. We know that if A * = ∅, then it is the unique minimal ideal of S which is called kernel of S.If S is an ordered semigroup and H ⊆ S, we denote(H] := {t ∈ S | t ≤ h for some h ∈ H}.[H) := {t ∈ S | t ≥ h for someh ∈ H}. Lemma 1. [2]If S is a commutative ordered semigroup, then prime ideals and weakly prime ideals coincide.

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A necessary condition is presented for the ideal class group of an imaginary quadratic function field K = k ( FD ) ( k = F, ( x ), 2T q ) to be of exponent < 2 . The condition is proved to be sufficient in some cases. An analogue of Louboutin's result in function field case is particularly presented.Keywords: imaginary quadratic function field, ideal class group, ideal class number. LOU BOUT IN[^]presented a necessary condition of Frobenius-Rabinovitsh style for the ideal class group of an imaginary quadratic number field K = Q ( F d ) (d$l(mod 8)) to be exponent <2 in terms of the primality of the values of a certain quadratic polynomial on consecutive integers. In this note we will give an analogue of this result for function field case. In fact, we present a more general result which implies the analogue as a particular case.A systematic research for quadratic function fields with odd characteristic was initiated by ~r t i n '~] who gave an analytic formula of the ideal class number by using the Gauss's method on binary quadratic forms. Let k = F q ( x ) be the rational function field over a finite field F, with odd q , D = D ( x ) a square-free polynomial in Fq [ x ] with degree d 21, sgn D the leading coefficient of D . The quadratic function field K = k ( r~ ) is called imaginary (by Artin) if 6 k , = Fq ( ($ ) ) ( the completion of k at the infinite prime divisor = ($)) . Without loosing of generality, we can assume sgn( D ) = 1 or g where g is a fixed generator of the cyclic group F," . The K is imaginary if and only if (type I : 2 l d ; or (type li ) : 21d and sgn D = g . Let OK be the integral closure of F, [ x I in K , C( OK) the ideal class group of OK, h ( OK) = I C ( O K ) I the ideal class number of O K . The 2-rank of the finite abelian group C ( O K ) has been determined by Zhang13'. We will state Zhang' s result and method in sec. 1. It is easy to see that the exponent exp( C( OK ) ) of C ( OK ) is <2 if and only if C ( OK ) is an elementary 2-group which is also equivalent to h ( OK) = 2 r . Otherwise h ( O K ) > 2 r . In sec. 2 we will present a necessary condition of exp( C( O K ) )<2 for imaginary quadratic function field K = k ( T D ) with r = s -1 where s is the number of monic irreducible factors of D = D ( x ) . The condition is also sufficient in some cases.