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Galois closures of commutative rank n ring extensions were introduced byBhargava and the second author. In this paper, we generalize the constructionto the case of non-commutative rings. We show that non-commutative Galoisclosures commute with base change and satisfy a product formula. As anapplication, we give a uniform construction of many of the representationsarising in arithmetic invariant theory, including many Vinberg representations.

Galois closures of non-commutative rings and an application to Hermitian representations