In this work, we present a standard model for Galois rings based on the standard model of their residual fields, that is, a sequence of Galois rings starting with ${mathbb Z}_{p^r} that coves all the Galois rings with that characteristic ring and suc
h that there is an algorithm producing each member of the sequence whose input is the size of the required ring.
A finite semifield $D$ is a finite nonassociative ring with identity such that the set $D^*=Dsetminus{0}$ is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which completes the cl
assification of finite semifields of order 125 or less.
A finite semifield is a finite nonassociative ring with identity such that the set of its nonzero elements is closed under the product. From any finite semifield a projective plane can be constructed. In this paper we obtain new semifield planes of o
rders 81 by means of computational methods. These computer-assisted results yield to a complete classification (up to isotopy) of 81-element finite semifields.
