Character Sums overp-adic Fields

Primitive Polynomials With Prescribed Second Coefficient

“…This follows from Corollary 6.1 of [21]. The significant point is that the polynomial (α 0 + 2α 1 )x ∈ R * 1,2 [x] has weighted degree 2 (if α 0 = 0) or 1 (if α 0 = 0).…”

“…We string the 82 pairs (a, c), one example for each primitive quartic (2,14), (3,5), (4,35), (5,14), (6,24), (7,20), (8,34), (9,14), (10, 2), (11,2), (12,8), (13,15), (14,6), (15,14), (16,80), (17,6), (18,32), (19,19), (20,43), (21,15), (22, 32), (23,32), (24,8), (25,62) In summary, the above examples suffice to complete the proof (for odd q) of Theorem 1.2 for n = 4, m = 2 and a = 0.…”

$p$ -ADIC FORMAL SERIES AND COHENS PROBLEM This work was supported by NSF of China with contract No. 19971096 and No. 90104035.

“…See for example [13], [15], etc. Here we give two theorems from [15] in a slightly different form for our later use. These results are analogous to Weil estimates on character sums over finite fields.…”

“…THEOREM 8. [15]. Let f (x) ∈ R e,nk [x] be nondegenerate with weighted e-degree D e (f (x)), ψ e,n a nontrivial additive character of R e,nk and χ a nontrivial multiplicative character of * nk .…”

“…In this paper, we shall transfer the working to the unramified extensions of the p-adic fields and their completions, as well as the appropriate quotient rings, Galois rings so that we can translate the existence of primitive polynomials into the existence of the primitive element solutions of Teichmüller points of some system of trace equations with each equation over a suitable Galois ring. To estimate the number of such primitive element solutions, we need the estimates of characters sums over Galois rings [13] and over the Teichmüller points of p-adic number fields [15].…”