This paper is concerned with the following four conjectures about the finite field \(\mathbb F_q\) which were made by \textit{S. W. Golomb} in 1984 [J. Comb. Theory, Ser. A 37, 13--21 (1984; Zbl 0547.05020)]. Conjecture A: If \(q>2\), then \(\mathbb F_q\) contains primitive elements \(\alpha\) and \(\beta\) with \(\alpha+\beta=1\). Conjecture B: If \(q>3\), then \(\mathbb F_q\) contains primitive elements \(\alpha\) and \(\beta\) with \(\alpha+\beta=-1\). Conjecture C: There exists an integer \(q_0\) such that for all \(q\geq q_0\), every nonzero element \(\varepsilon\) of \(\mathbb F_q\) has at least one representation as \(\varepsilon=\alpha+\beta\) where \(\alpha\) and \(\beta\) are primitive elements of \(\mathbb F_q\). Conjecture D: There exists a primitive quadratic of trace 1 over \(\mathbb F_q\) for every \(\mathbb F_q\). With the present paper, all of these conjectures have now been shown to be true. The paper first surveys the work on Golomb's conjectures, pointing out the status of each one and it concludes with a complete resolution of the remaining conjectures. In addition to theoretical proofs, which apply when \(q\) is sufficiently large, the paper reports on computer studies which were used to resolve small values of \(q\). As a consequence of the work, the existence of several conjectured classes of Costas arrays are now known to exist. (A Costas array of order \(n\) is an \(n\times n\) permutation matrix with the property that the \(({n\atop 2})\) vectors obtained by connecting each pair of ones in the matrix are all different). \{In an Erratum published in the same journal 2, No. 4, 297--299 (1992) the corrected Table 2 (pp. 50--52 in the original) is given \}.