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Rings and Modules Seminar
~ Abstracts ~

R. Padmanabhan
padman(at)cc(dot)umanitoba(dot)ca

Department of Mathematics
University of Manitoba

Tuesday, September 28, 2010

Axiom Systems for Fields
Abstract:

Sometime during the last summer, Dr. Tommy Kucera brought to my attention a rather remarkable set of axioms for fields discovered by G. Pickert. Instead of the usual two binary operations of addition and multiplication connected by the distributive laws, PIckert's axioms demand the existence of three group laws with apparently no connecting laws like distributivity. Pickert's third group operation is the now familiar "circle operation" of rings defined by

x•y = x+y-xy.
This happens to be a special case of the ternary law of composition
d(x,y,z) = xz+y-xy.
In fact, d(x,y,1) = x•y. It is known (see [2]) that fields can be defined with this single ternary operation (analogous to the so-called ternary Boolean algebras). These axiom systems arose in connection with characterizing the class of all groups which are multiplicative groups of fields. In this presentation, I will mention three such axiom systems and will go through the proof of at least one of them.

  1. P.M. Cohn, Skew Fields, Cambridge University Press, 2003.
  2. R.M. Dicker, A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field. Proc. London Math. Soc. (3) 18, 1968, 114–124.
  3. W. Leissner, Eine Charakterisierung der multiplikativen Gruppe eines Körpers, Jber.Deutsch. Math.-Verein. 73, 92–100.
  4. G. Pickert, Eine Kennzeichnung desarguesscher, Ebenen. Math. Z. 71, 1959, 99– 108.

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