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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
##### Keywords:
Semifields; division rings; finite geometries
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