On Sums and Products in a Field

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 + … + a_k with a₁, …, _ak ∈ F and a₁… a_k = 1, and that for any α ∈ F {0} we can write every x ∈ F as a₁ … a_k with a₁, …, a_k ∈ F and a₁ + … + a_k = α. We also prove that for any x ∈ F and k ∈ {2, 3, …} there are a₁, …, a_2k ∈ F such that a₁ + … + a_2k = x = a₁ … a_2k.