Let F(z) be a polynomial with coefficients in a perfect field k and let K be the normal closure of k(z) over k(F). All polynomials for which the genus of K over k is one are determined; they depend in part on the characteristic of k. Some results for higher genus are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 12 F 10.

Let p,m be the modular representation algebra of the cyclic group of order pm over the prime field Zp. The characters of p, m are derived. For p = 2, this provides an alternative proof of a result due to Carlson (1975), tha 2,m is a local ring. It is shown that for p>2, p, m is a direct sum of 2m local rings. Their dimensions and primitive idempotents are derived.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20, 12 C 05, 12 C 30, 33 A 65.

Let Bp be the bicyclic semigroup over P=G⋒[1,∞ where G is a subgroup of the multiplicative group of positive real numbers. If + is an addition which makes Bp, with its usual multiplication, into a semiring, then + is idempotent, and P is embedded as a sub-semiring in Bp and for each x in p, 1≦x+1≦x and 1≦1+x≦x. We show that any idempotent addition on P with these inequalities holding is max, min or trivial. The trivial addition on P extends trivially. If addition on P is min, then let , and . We charactertize all additions on Bp in terms of U and U′; and, in particular If U=U′ is a proper subset of R1 we demonstrate a correspondence between all such additions and certain homomorphisms of G to (0,∞)

Subject classification (Amer. math. Soc. (MOS) 1970): primary 16 A 80; secondary 22 A 15.

In this paper, the author generalizes the concept of thinness introduced by Kingman and Robertson (1968) to study the convexity of the range of a multimeasure. It is proved that every thin multimeasure taking values in a Fréchet space has convex range, and that, for a suitable measureable space, if a multimeasure is non-atomic, then the weak closure of its range is convex.

If U and V are toplogies on an abstract set x, then the triple (X, U, V) is a bitopologica space. Using the theorem of Priestley on the representation of distributive lattices, results of Dilworth concerning the normal completion of the lattice of bounded, continuous, realvalued functions on a topological space are extended to include the lattice of bounded, semi-continuous, real-valued functions on certain bitopological spaces. The distributivity of certain lattices is investigated, and the theorem of Funayama on distributive normal completions is generalized.

A ring R for which every finitely generated right submodule of SR, the left flat epimorphic hull of R, is projective is termed an extended semi-hereditary ring. It is shown that several of the characterizing properties of Prufer domains may be generalized to give characterizations of extended semi-hereditary rings. A suitable class of PP rings is introduced which in this case serves as a generalization of commutative integral domains.

We investigate conditions on a permutation group G sufficient to ensure that G fixes a graph in any switching class of graphs that it stabilizes. Our main result gives a necessary and sufficient condition for a dihedral group G to have this property.

The divisor density of an integer sequence is defined: this measures the tendency of divisors of (almost all) integers to belong to the sequence. A proof of a conjecture of Erdös is given and this is linked to some previous conjectures of the author's concerning distribution (mod 1) of functions of the divisors.

In Russell (1973) a Riemann-type necessary and sufficient condition was given for the existence of (defined also in Russell (1975)) when f was bounded and g was k-convex in [a′, b′]. In this paper we present necessary and sufficient conditions for the existence of a particular Stieltjes-type integral without imposing a convexity condition upon g. These conditions are used to obtain an additivity result for the integral over adjoining intervals without any additional restrictions being imposed upon the functions involved.