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Let $\mathbb{F}_q$ be a finite field with $q$ elements. M. Gerstenhaber andIrving Reiner has given two different methods to show the number of matriceswith a given characteristic polynomial. In this talk, we will give anotherproof for the particular case where the characteristic polynomial isirreducible. The number of such matrices is important to know the efficiency ofan algorithm to factor polynomials using Drinfeld modules.

The number of matrices over $\mathbb{F}_q$ with irreducible characteristic polynomial