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Distribution of constant terms of irreducible polynomials in Zp[x]

Sarah C. Cobb, Department of Mathematics, Midwestern State University, Michelle L. Knox, Marcos Lopez, Terry McDonald, Patrick Mitchell, Department of Mathematics, Midwestern State University, Department of Mathematics, Midwestern State University, Department of Mathematics, Midwestern State University, Department of Mathematics, Midwestern State University
2019 Notes on Number Theory and Discrete Mathematics  
We obtain explicit formulas for the number of monic irreducible polynomials with prescribed constant term and degree q k over a finite field. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a residue in the underlying field. We further show that as k becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal.
doi:10.7546/nntdm.2019.25.4.72-82 fatcat:fnm3tkr6enfrhisznjhx2xpp3a