  Given a simply connected space $X$ with the cohomology $H^*(X;{\mathbb Z}_2)$
to be polynomial, we calculate the loop cohomology algebra $H^*(\Omega
X;{\mathbb Z}_2)$ by means of the action of the Steenrod cohomology operation
$Sq_1$ on $H^*(X;{\mathbb Z}_2).$ As a consequence we obtain that $H^*(\Omega
X;{\mathbb Z}_2)$ is the exterior algebra if and only if $Sq_1$ is
multiplicatively decomposable on $H^{\ast}(X;{\mathbb Z}_2).$ The last
statement in fact contains a converse of a theorem of A. Borel.

Algebraic Topology