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# Algebra Seminar

Date/Time/Room: Friday (12/05/2005) at 4:00pm in 487 Pickard Hall

Speaker: Bill Purpura, Department of Mathematics, University of Texas at Arlington

### "Counting the Generalized Twisted Fields"

Abstract: In 1961, the algebraist A.A. Albert discovered a semifield known as a generalized twisted field. Given a finite field F=GF(p^n), distinct non-identity automorphisms \alpha and \beta of F, and an element c in F satisfying a certain property, one can define a new product \circ on F by "twisting" the multiplication of F: x \circ y =xy-c\alpha(x)\beta(y) This gives rise to a generalized twisted field. In this talk we exhibit a procedure to count the number of non-isotopic generalized twisted fields g(p^n) for a given order. We show that g(p^n) is a polynomial in p, and for fixed p has the remarkable property that it is dependent not on the magnitude of n, but rather on the prime factorization of n. This talk will be accessible to anyone with a basic knowledge of finite fields and modular arithmetic.

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