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Let $A$ be a unital associative algebra over a field $k$. All unitalassociative algebras containing $A$ as a subalgebra of a given codimension$\mathfrak{c}$ are described and classified. For a fixed vector space $V$ ofdimension $\mathfrak{c}$, two non-abelian cohomological type objects areexplicitly constructed: ${\mathcal A}{\mathcal H}^{2}_{A} \, (V, \, A)$ willclassify all such algebras up to an isomorphism that stabilizes $A$ while${\mathcal A}{\mathcal H}^{2} \, (V, \, A)$ provides the classification fromH\"{o}lder's extension problem viewpoint. A new product, called the unifiedproduct, is introduced as a tool of our approach. The classical crossed productor the twisted tensor product of algebras are special cases of the unifiedproduct. Two main applications are given: the Galois group ${\rm Gal} \, (B/A)$of an extension $A \subseteq B$ of associative algebras is explicitly describedas a subgroup of a semidirect product of groups ${\rm GL}_k (V) \rtimes {\rmHom}_k (V, \, A)$, where the vector space $V$ is a complement of $A$ in $B$.The second application refers to supersolvable algebras introduced as theassociative algebra counterpart of supersolvable Lie algebras. Several explicitexamples are given for supersolvable algebras over an arbitrary base field,including those of characteristic two whose difficulty is illustrated.Comment: 30 pages; new version: title changed; added a new section on Galois  extensions of associative algebras. Final version to appear in Monatsh. fur  Mathematik. DOI:10.1007/s00605-015-0814-

Extending structures, Galois groups and supersolvable associative algebras