Tame and wild two by two matrix orders (Q1314264)

For a field \(K\) with a complete discrete valuation and valuation ring \(D\), the tame \(D\)-orders in the second order matrix algebra \(M_ 2(K)\) are classified. The authors give an exact definition for tame and wild orders in general, and prove that tame-wild dichotomy holds in this particular case. They show that an order in \(M_ 2(K)\) is tame if and only if it is not contained in a conjugate of some particular wild order. An explicit description of the tame orders in \(M_ 2(K)\) is included.