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

The loop cohomology of a space with the polynomial cohomology algebra