Is there any good reference for the Pontrjagin ring structure on 
$$
H\_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H\_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)?
$$
I am familiar with Serre's theorem describing the mod 2 cohomology ring structure. I'm also aware that the action of the Dyer--Lashof operations on this infinite loop space is trivial. 



Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces
By naturality and the external Cartan formula, the standard polynomial generators of $H^\*(K(\mathbf{Z}/2,k);\mathbf{Z}/2)$ given by iterated Steenrod operations on the fundamental class are primitive. Therefore the homology is a divided power algebra on the dual elements.


