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In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4.〔Although the constructions will provide a projective plane of order 4, the unique such plane is Desarguesian and is generally not considered to be a Hall plane.〕 ==Algebraic construction via Hall systems== The original construction of Hall planes was based on a Hall quasifield (also called a ''Hall system''), H of order ''p''2''n'' for ''p'' a prime. The construction of the plane is the standard construction based on a quasifield (see Quasifield#Projective planes for the details.). To build a Hall quasifield, start with a Galois field, for ''p'' a prime and a quadratic irreducible polynomial over ''F''. Extend ''H'' = ''F'' × ''F'', a two-dimensional vector space over ''F'', to a quasifield by defining a multiplication on the vectors by when and otherwise. Writing the elements of ''H'' in terms of a basis <1, λ>, that is, identifying (''x'',''y'') with ''x'' + λ''y'' as ''x'' and ''y'' vary over ''F'', we can identify the elements of ''F'' as the ordered pairs (''x'', 0), i.e. ''x'' + λ0. The properties of the defined multiplication which turn the right vector space ''H'' into a quasifield are: # every element α of ''H'' not in ''F'' satisfies the quadratic equation f(α) = 0; # ''F'' is in the kernel of ''H'' (meaning that (α + β)c = αc + βc, and (αβ)c = α(βc) for all α, β in ''H'' and all c in ''F''); and # every element of ''F'' commutes (multiplicatively) with all the elements of ''H''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hall plane」の詳細全文を読む スポンサード リンク
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