Factoring Polynomials in B[x]

By Dr. Francisco Alarcón
Professor
Mathematics Department
IUP

The finite semifield B={0,1} contains the same elements as Z2 (the integers mod 2) but in B we have 1+1=1. Amazingly this is the only finite semifield that is not a field. Since Z2 and B have the same elements, the sets of polynomials in the indeterminate x are the same for both of these structures. However the operations on the polynomials are not the same.

We will consider polynomials over both the field Z2 and the semifield B. We will consider the problem of factoring polynomials over both of these structures. While factorizations over Z2 have been studied, the factorizations of polynomials over B have not been studied. Some interesting results about zero divisors and cancellation laws will be presented along with some other preliminary results.

While some knowledge of Abstract Algebra would be useful, it is not necessary for following most of the material presented.



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