Abstract Let 𝔽 q {\mathbb{F}_{q}} be the finite field of q = p k {q=p^{k}} elements with p being a prime and let k be a positive integer. For any y , z ∈ 𝔽 q {y,z\in\mathbb{F}_{q}} , let N s ( z ) {N_{s}(z)} and T s ( y ) {T_{s}(y)} denote the numbers of zeros of x 1 3 + ⋯ + x s 3 = z {x_{1}^{3}+\cdots+x_{s}^{3}=z} and x 1 3 + ⋯ + x s - 1 3 + y x s 3 = 0 {x_{1}^{3}+\cdots+x_{s-1}^{3}+yx_{s}^{3}=0} , respectively. Gauss proved that if q = p {q=p} , p ≡ 1 ( mod 3 ) {p\equiv 1~{}(\bmod~{}3)} and y is non-cubic, then T 3 ( y ) = p 2 + 1 2 ( p - 1 ) ( - c + 9 d ) , T_{3}(y)=p^{2}+\frac{1}{2}(p-1)(-c+9d), where c and d are uniquely determined by 4 p = c 2 + 27 d 2 {4p=c^{2}+27d^{2}} and c ≡ 1 ( mod 3 ) {c\equiv 1~{}(\bmod~{}3)} except for the sign of d. In 1978, Chowla, Cowles and Cowles determined the sign of d for the case of 2 being a non-cubic element of 𝔽 p {\mathbb{F}_{p}} . But the sign problem is kept open for the remaining case of 2 being cubic in 𝔽 p {\mathbb{F}_{p}} . In this paper, we solve this sign problem by determining the sign of d when 2 is cubic in 𝔽 p {\mathbb{F}_{p}} . Furthermore, we show that the generating functions ∑ s = 1 ∞ N s ( z ) x s {\sum_{s=1}^{\infty}N_{s}(z)x^{s}} and ∑ s = 1 ∞ T s ( y ) x s {\sum_{s=1}^{\infty}T_{s}(y)x^{s}} are rational functions for any z , y ∈ 𝔽 q * := 𝔽 q ∖ { 0 } {z,y\in\mathbb{F}_{q}^{*}:=\mathbb{F}_{q}\setminus\{0\}} with y being non-cubic over 𝔽 q {\mathbb{F}_{q}} , and we also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.