Related papers: Finite Hyperfields of Order $n\leq 5$
We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients…
We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the…
The concern of this paper is to show that there always exist Krasner hyperfields of order n, where n is an integer greater than or equal to 2.
We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…
We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.
We construct explicitly in any finite field of the form Fq[x]/(x^m-a) elements with multiplicative order at least 2^{(2m)^(1/2)}
The quotient hyperfield is a landmark on the borderline of fields and hyperfields. In this paper, which is the second part of our previously published paper, all the hyperfields of order 7 are constructed, enumerated and presented, in the…
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…
In this paper we prove that there are exactly eight function fields, up to isomorphism, over finite fields with class number one.
In this paper, we describe finite, additively commutative, congruence simple semirings. The main result is that the only such semirings are those of order 2, zero-multiplication rings of prime order, matrix rings over finite fields, those…
We use fast-growing finite and infinite sequences of natural numbers and more complicated constructs to define models of hypercomputation and interpret non-arithmetic predicates, with the strongest extensions reaching full second order…
A finite semifield $D$ is a finite nonassociative ring with identity such that the set $D^*=D\setminus\{0\}$ is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which…
The first part is expository: it explains how finite fields may be used to prove theorems on infinite fields by a reduction mod p process. The second part gives a variant of P.Smith's fixed point theorem which applies in any characteristic.
In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \leq 5$. Further we give some connections…
We give sufficient conditions to find all subtypes isomorphic to a subtype in a finite generalized ordered type.
We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs…
We consider the dimensions of finite type of representations of a partially ordered set, i.e. such that there is only finitely many isomorphism classes of representations of this dimension. We give a criterion for a dimension to be of…
We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…
In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.
Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.