# Abstracts of Papers

#### Abstract

S OF PAPERS SUBMITTED FOR PRESENTATION TO THE SOCIETY The following papers have been submitted to the Secretary and the Associate Secretaries of the Society for presentation at meetings of the Society. They are numbered serially throughout this volume. Cross references to them in the reports of the meetings will give the number of this volume, the number of this issue, and the serial number of the abstract. ALGEBRA AND THEORY OF NUMBERS 299. Leonard Carlitz: Representation of a polynomial in certain forms. Explicit formulas are found for the representation of an irreducible polynomial P in GF(p) in certain factorable forms of degree k in k indeterminates. For example, when k — 2, p5*2, the form may be taken as t/ —«F , where a. is not a square in GF(p). Here U is determined by the formula 2A(x)^(-l)[l][3] • • • [2h-l] (mod P(x)), where P is of degree 2h, and [i] =x*>—x (Duke Mathematical Journal, vol. 9 (1942), pp. 234-243, p. 242). Similar results are obtained for the general case. (Received August 3, 1942.) 300. Leonard Carlitz: Some formulas f or the composition of numerical functions. This note is concerned with the sum h(M)=J^f{U)g(V) extended over polynomials of fixed degree k in GF(p) such that (a+p)M=aU+(3V, a, j8 in GF{p); it is assumed that f(U) and g(V) are of the form ^2,e{D), where D runs through the divisors of U of degree a t most k/2. I t is found that h{M) is of the same general form. In various special cases h(M) can be expressed in quite simple form. In particular the formulas of the present paper include formulas occurring in the problem of representing a polynomial as the sum of an even number of squares. (Received August 3, 1942.) 301. R. P. Dilworth: On the decomposition theory of modular lattices. The Kurosch-Ore decomposition theorem asserts that if an element a of a modular lattice has two reduced decompositions into irreducibles a=p\C\ • • • C\pm = qiC\ • • • r\gn, then each pi may be replaced by a suitably chosen q7-. However, this leaves unanswered questions like the following: Does each #/ replace some pi? Can different p's be replaced by different g's? The following precise result is proved: If a—p\C\ • • • r\pm — gi(^ ' r\qn, then the q's can be renumbered in such a way that each pi may be replaced by g*. This theorem requires a much deeper analysis of the arithmetical structure of a modular lattice. For this purpose extensive use is made of the concept of an over-divisor which is defined as follows: a is said to overdivide b, or to be an over-divisor of b if xC\a = b implies x = b. (Received August 4, 1942.)