Asymptotic formula for the number of irrreducible trinomials divisible by a given irrreducible polynomial (Q1280389)

Let \(f(z)\) be a polynomial of degree \(n\), irreducible over \(GF(p)\) and \(g\) be a primitive element of the multiplicative group \(GF(p^\alpha)\) which is the root of \(f(z)\). \(T\) denotes the number of solutions to the equation \[ y = \text{ind} (g^x-1),\quad x=1,\dots,M;\quad y=1,\dots,N. \] The main result of the paper is as follows. If the polynomial \(f(z)\) of degree \(n\) is irreducible over \(GF(p)\), \(f(g)=0\), then \[ T=MN/(P-1)+O(\sqrt{P} \ln^2P), \] where \(P=p^\alpha\).