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.
Available by permission of the author. Reproduction or retransmission of this material in any form is prohibited without expressed written permission of the author.
Holbrook, Brian, "Irreducible Polynomials over a Finite Field Zp" (1994). Honors Theses, 1963-2015. 474.