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For any positive integers we present formulae for the number of irreducible polynomials of degree over the finite field where the coefficients of , and are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period in .
Thomas Mountford, Michael Cranston