The Novikov-Landweber algebra and theSteenrod algebra are set up in terms of the primitive differential operators$D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way ontheintegralpolynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. Acommutative wedge product$\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\veer}$is divisible by $r!$ as an integraloperator. Thedivided differential operator algebra $\cal D$ is generated over the integers by thedividedoperators $\frac{D_k^{\veer}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic tothe abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect tothe composition and wedge products.Under composition$\cal D$ is isomorphic to the Landweber-Novikov algebra. AHopf sub-algebrais generated under compositionby the integral Steenrod squares$SQ^r = \frac{D_1^{\veer}}{r!}$ and reducesmod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.