Galois Closures of Non-commutative Rings and an Application to Hermitian Representations

Galois closures of commutative rank $n$ ring extensions were introduced by Bhargava and the 2nd author. In this paper, we generalize the construction to the case of non-commutative rings. We show that noncommutative Galois closures commute with base change and satisfy a product formula. As an application, we give a uniform construction of many of the representations arising in arithmetic invariant theory, including many Vinberg representations.