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A survey of finite semifields. (English) Zbl 1031.12009
An algebraic system \(S\) with two binary operations (addition and multiplication) is called a semifield if the following axioms are fulfilled:
1. \((S;+)\) is a group with identity \(0\).
2. If \(a,b\in S\) and \(ab=0\) then \(a=0\) or \(b=0\).
3. If \(a,b,c\in S\) then \(a(b+c)=ab+ac\) and \((a+b)c=ac+bc\).
4. There exists an \(e\in S\) satisfying the relationship \(ea=ae=a\) for all \(a\in S\).
The aim of this paper is the brief observation of \(101\) references about finite semifields.
In the first part a classification of semifields of finite order is adduced.
The second part is a catalogue of the known semifields of finite order.

MSC:
12K10 Semifields
17A35 Nonassociative division algebras
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