It was shown by \textit{Y. Felix}, \textit{S. Halperin}, \textit{C. Jacobsson}, \textit{C. Löfwall}, and \textit{J.-C. Thomas} [Am. J. Math. 110, 301-322 (1988; Zbl 0654.55011)] that the rational Lusternik-Schnirelmann category of a space \(X\) forms an upper bound for the depth of the rational homology algebra of the loop space \(\Omega\) X. The present work extends this result to the case of an arbitrary coefficient field \(\Bbbk\). As in the rational context, the proof relies on techniques of differential algebra. However, unlike the rational situation, characteristic \(p>0\) necessitates modeling the non-commutative DGA of singular \(\Bbbk\)-cochains and proving a non-commutative version of Theorem B of the paper cited above (Theorem A'). The \(\Bbbk\)-depth theorem above is applied, together with structure results on radicals of Hopf algebras and facts about Gorenstein algebras and spaces [see \textit{Y. Felix}, \textit{S. Halperin} and \textit{J.-C. Thomas}, Adv. Math. 71, No.1, 92-112 (1988; Zbl 0659.57011)], to prove the following interesting generalization of Serre's theorem on the existence of infinitely many nontrivial homotopy groups: Let \(X\) be simply connected with finite category and suppose that for some prime \(p>0\) each \(H_ i(X;{\mathbb{Z}}/p)\) is finite dimensional. If \(X\) admits an \(n\)-stage Postnikov decomposition at \(p\), then \(H_ i(X:{\mathbb{Z}}/p)=0\) for all \(i\geq 1\). This result says (for \(p\)-local \(X\) say) that noncontractibility not only implies the existence of infinitely many homotopy groups, but that these groups in fact require an infinite number of principal fibrations to assemble \(X\).