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 such 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 classification 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 orders 81 by means of computational methods. These computer-assisted results yield to a complete classification (up to isotopy) of 81-element finite semifields.
