A strongly regular graph is a graph in which each vertex has the degree k, any two adjacent vertices have \(\lambda\) common neighbours and any two non-adjacent vertices have \(\mu\) common neighbours, where k, \(\lambda\), \(\mu\) are fixed numbers. In the paper the decomposition of a complete graph into three strongly regular graphs is studied. A theorem on the number of vertices n and on the numbers k, \(\lambda\), \(\mu\) is proved. Applications in the theory of fields are considered. A theorem is proved which concerns the number of non-trivial solutions (x,y,z) of the equation \(x^ 3+y^ 3=z^ 3\) in a finite field of order \(p^{2h}\), where p is a prime number congruent with 2 modulo 3.