I am reading through the Dyer and Lashof's paper "Homology of Iterated Loop Spaces". They are quoting the following theorem:
If $A$ is a connected, free, associative, commutative, primitively generated mod $p$ Hopf algebra of finite type, then $A$ is isomorphic as a Hopf algebra to a tensor product of free monogenic Hopf algebras, i.e. to a tensor product of exterior and polynomial Hopf algebras each having one generator.
I can find respective theorems in Milnor-Moore's paper, but I cannot understand the "freeness" assumption, as it does not appear elsewhere.
So my questions are: 1) What "free Hopf algebra" might mean in this context? Is it free as an algebra?
2) Is it known that the only "free" Hopf algebras with one generator are external and polynomial ones?