The loop cohomology of a space with the polynomial cohomology algebra

Given a simply connected space X with polynomial cohomology H∗(X;Z2), we calculate the loop cohomology algebra H∗(ΩX;Z2) by means of the action of the Steenrod cohomology operation Sq1 on H∗(X;Z2). This calculation uses an explicit construction of the minimal Hirsch filtered model of the cochain algebra C∗(X;Z2). As a consequence we obtain that H∗(ΩX;Z2) is the exterior algebra if and only if Sq1 is multiplicatively decomposable on H∗(X;Z2). The last statement in fact contains a converse of a theorem of A. Borel (Switzer, 1975, Theorem 15.60).