Affiliation: | 1.Department of Mathematics, Nanjing University, Gulou Campus, No. 22, Hankou Road, Gulou District, Nanjing, 210093, P. R. China ; |
Abstract: | We study sums and products in a field. Let F be a field with ch(F) ≠ 2, where ch(F) is the characteristic of F. For any integer k ? 4, we show that any x ∈ F can be written as a1 + … + ak with a1, …, ak ∈ F and a1… ak = 1, and that for any α ∈ F {0} we can write every x ∈ F as a1 … ak with a1, …, ak ∈ F and a1 + … + ak = α. We also prove that for any x ∈ F and k ∈ {2, 3, …} there are a1, …, a2k ∈ F such that a1 + … + a2k = x = a1 … a2k. |