The order of a typical matrix with entries in a finite field

Abstract

IfA is an invertiblen×n matrix with entries in the finite field Fq, letT n (A) be its minimum period or exponent, i.e. its order as an element of the general linear group GL(n,q). The main result is, roughly, that\(T_n (A) = q^{n - } (log n)^{2 + 0(1)} \) for almost everyA.

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Supported by N.S.F. (D.M.S. 9101753).

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Schmutz, E. The order of a typical matrix with entries in a finite field. Israel J. Math. 91, 349–371 (1995). https://doi.org/10.1007/BF02761656

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Keywords

  • Finite Field
  • Characteristic Polynomial
  • Moment Generate Function
  • Irreducible Polynomial
  • Typical Matrix