The Ubiquity of the Orders of Fractional Semifields of Even Characteristic

To each non-square integer \(2^{2N+1}\ge 2^5\) there correspond semifields \(D\) of order of \(2^{2N+1}\) that contain \(\text{ GF}(4)\). Hence there exist affine planes for each non-square order \(2^{2N+1}\ge 2^{5}\) that contain subaffine planes of order \(2^2\). Moreover, there also exists semifields \(D_1\) and \(D_2\), with \(|D_1|= |D_2| =|D|\) such that \(D_1\) is commutative and \(D_2\) is non-commutative but neither \(D_1\) nor \(D_2\) contains \(\text{ GF}(4)\).

1. The procedure fails for strict presemifields \((F,+, \bullet )\) whenever \(\mathcal F \) does not include the identity map on \(F\).

The author is deeply indebted to both the referees for their meticulous perusal of the initial version of this paper and especially for detecting a crucial error in a lemma involved in the proof of the ‘ubiquity’ of non-fractional non-commutative semifields (theorem B); I am equally indebted to Prof. Linlin Chen for pointing out this error. The present version of the paper includes an entirely new proof of theorem B, which has also led to a much simplified proof of the main result, the ubiquity of fractional semifields, theorem A.

Authors and Affiliations

1. 2 Marchmont Terrace, Glasgow, G12 9LT , United Kingdom

   Vikram Jha

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Correspondence to Vikram Jha.

Communicated by S. Ball.

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