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Existence Problems of Primitive Polynomials over Finite Fields
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This book concerns existence of primitive polynomials over finite fields with an arbitrarily prescribed coefficient. It completes the proof of a fundamental conjecture of Tom Hansen and Gary L. Mullen (1992) which asserts that, with some explicablegeneral exceptions, there always exists a primitive polynomial ofa ny degree over any finite field with an arbitrary coefficient prescribed. Here, the last remaining cases of the conjecture are proven efficiently, in a self-contained way and with very little computation. This is achieved by separately considering thepolynomials with second, third or fourth coefficient prescribed,and in each case developing methods involving the use of charactersums and sieving techniques. When the characteristic of the fieldis 2 or 3, p-adic analysis is used. The book also researches theexistence of primitive polynomials with two coefficients prescribed (the constant term and any other coefficient).
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