Minimal proper quasifields with additional conditions. (English) Zbl 1534.12003
Summary: We investigate the finite semifields which are distributive quasifields, and finite near-fields which are associative quasifields. A quasifield \(Q\) is said to be a minimal proper quasifield if any of its sub-quasifield \(H\ne Q\) is a subfield. It turns out that there exists a minimal proper near-field such that its multiplicative group is a Miller-Moreno group. We obtain an algorithm for constructing a minimal proper near-field with the number of maximal subfields greater than fixed natural number. Thus, we find the answer to the question: Does there exist an integer \(N\) such that the number of maximal subfields in arbitrary finite near-field is less than \(N\)? We prove that any semifield of order \(p^4 (p\) be prime) is a minimal proper semifield.
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