In this paper, for positive integers m, M, and a prime p such that M|p ^m – 1, we derive linear complexity over the prime field F _p of M-ary Sidel’nikov sequences of period p ^m – 1 using discrete Fourier transform. As a special case, the linear complexity of the ternary Sidel’nikov sequence is presented. It turns out that the linear complexity of a ternary Sidel’nikov sequence with the symbol k ₀ ≠1 at the (p ^m –1)/2-th position is nearly close to the period of the sequence, while that with k ₀ =1 shows much lower value.