Generalized twisted fields

Albert, A. A.

doi:10.2140/pjm.1961.11.1

Cited by 102 publications

(134 citation statements)

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“…Such a system has various names in the literature, but the term semifield is due to Knuth [21]. Some examples of finite semifields relevant to the study of unitals are the Dickson semifields [9,10], Albert's twisted fields [2], and Albert's generalized twisted fields [3,4].…”

Section: Dickson Semifield Planes and Dickson-ganley Unitalsmentioning

confidence: 99%

Non-classical polar unitals in finite Dickson semifield planes

et al. 2013

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Abstract. We give three proofs, two intrinsic and one extrinsic, that every Dickson-Ganley unital U(σ), parametrized by a field automorphism σ, is non-classical if σ is not the identity, extending a result of Ganley's (Math Z 128:34-42, 1972); we prove that U(σ1) is isomorphic to U(σ2) if and only if σ1 = σ2 or σ1 = σ −1 2 ; and we determine the (design) automorphism group of U(σ) as the collineation subgroup of the ambient Dickson semifield plane stabilizing the unital. This contains as a special case the corresponding result of O'Nan's (J Algebra 20:495-511, 1965) on the classical unital. Mathematics Subject Classification (2000). 51A35, 05B05, 17A35, 51A10, 51A45.

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Section: Dickson Semifield Planes and Dickson-ganley Unitalsmentioning

confidence: 99%

Non-classical polar unitals in finite Dickson semifield planes

et al. 2013

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“…An important example of pre-semifields are the so-called generalised twisted fields (or (Albert) twisted fields) constructed by A. A. Albert in [4], where multiplication on F q n is defined by…”

Section: Theorem 14 ( [23])mentioning

confidence: 99%

Finite semifields and nonsingular tensors

Lavrauw

2012

Des. Codes Cryptogr.

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“…So examples of non-associative division algebras include the associative ones, the octonions over the real numbers, and the twisted fields of Albert, which he first introduced over a finite field. He later generalized his definition to what he called generalized twisted fields [3]. Menichetti [22] gave a definition of these over any field, which we state more generally here.…”

Section: Preliminaries On Non-associative Division Algebrasmentioning

confidence: 99%

“…The associative algebras are included among the non-associative ones, but the first example of such a division algebra that was not associative was the octonions of Graves and Cayley over the real numbers [8]. Dickson constructed examples of non-associative division algebras over other fields [14], [15], as did Albert, who systematized the subject [1], [2], [3]. Since a semifield is a non-associative division algebra over its center, these algebras have been much researched and are important for the study of translation planes and finite geometries (see [7] for details and references).…”

Section: Introductionmentioning

confidence: 99%