Irreducible Polynomials over a Finite Field Zp

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Robert Dumonceaux


The topic of my thesis was counting irreducible polynomials. I began with some preliminary material including relevant theorems discovered in various math textbooks and data generated by a computer program that I wrote. I then used this data to calculate formulas for the number of polynomials which had zeroes in a field Zp of sufficiently low degree. Next, I explained why it would not be practical to use the formulas for lower degrees to synthesize formulas for high degrees. However, I was able to generate a new formula using a different counting technique: the inclusion-exclusion principle. Finally, I discussed the possibilities of generalizing my findings to all finite fields or to commutative rings.

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