Existence problems of primitive polynomials over finite fields

This thesis concerns existence of primitive polynomials over finite fields with one coefficient arbitrarily prescribed. It completes the proof of a fundamental conjecture of Hansen and Mullen (1992), which asserts that, with some explicable general exceptions, there always exists a primitive polynomial of any degree n over any finite field with an arbitrary coefficient prescribed. This has been proved whenever n is greater than or equal to 9 or n is less than or equal to 3, but was unestablished for n = 4, 5, 6 and 8. In this work, we efficiently prove the remaining cases of the conjecture in a selfcontained way and with little computation; this is achieved by separately considering the polynomials with second, third or fourth coefficient prescribed, and in each case developing methods involving the use of character sums and sieving techniques. When the characteristic of the field is 2 or 3, we also use p-adic analysis. In addition to proving the previously unestablished cases of the conjecture, we also offer shorter and self-contained proof of the conjecture when the first coefficient of the polynomial is prescibed, and of some other cases where the proof involved a large amount of computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove the existence of primitive polynomials with two coefficients prescribed (the constant term and any other coefficient)