Lazard's theorem is a central result in formal group theory; it states that the ring over which the universal formal group law is defined (known as the Lazard ring) is a polynomial algebra over the integers with infinitely many generators. This ring also shows up in algebraic topology as the complex cobordism ring. The main aim of this paper is to show that the polynomial structure of the Lazard ring follows from the polynomial structure of a certain subalgebra of symmetric functions with integer coefficients. The connection between symmetric functions and the Lazard ring is provided by a certain Hopf algebra map from symmetric functions to the covariant bialgebra of a formal group law. We study this map by deriving formulas for the images of certain symmetric functions; in passing, we use this map to prove some symmetric function and Catalan number identities. Based on the above results, we prove Lazard's theorem, and present an application to the construction of certain p-typical formal group laws over the integers. Combinatorial methods play a major role throughout this paper.
N. Metropolis and G.-C. Rota [Adv. Math 50, 1983, 95-125] studied the necklace polynomials, and were lead to define the necklace algebra as a combinatorial model for the classical ring of Witt vectors (which corresponds to the multiplicative formal group law X+Y-XY). In this paper, we define and study a generalized necklace algebra, which is associated with an arbitrary formal group law F over a torsion free ring A. The map from the ring of Witt vectors associated with F to the necklace algebra is constructed in terms of certain generalizations of the necklace polynomials. We present a combinatorial interpretation for these polynomials in terms of words on a given alphabet. The actions of the Verschiebung and Frobenius operators, as well as of the p-typification idempotent are described and interpreted combinatorially. A q-analogue and other generalizations of the cyclotomic identity are also presented. In general, the necklace algebra can only be defined over the rationalization of A. Nevertheless, we show that for the family of formal group laws over the integers F_q(X,Y)=X+Y-qXY, q in Z, we can define the corresponding necklace algebras over Z. We classify these algebras, and define isomorphic ring structures on the groups of Witt vectors and the groups of curves associated with the formal group laws F_q. The q-necklace polynomials, which turn out to be numerical polynomials in two variables, can be interpreted combinatorially in terms of so-called q-words, and they satisfy an identity generalizing a classical one.