Polyphase versions of the well–known binary Legendre or quadratic–residue sequences were first described and analysed by Sidelnikov over thirty years ago, but have since received very little attention. Here, it is shown that these q–phase sequences of prime length L can also be constructed from the index sequence of length L or, equivalently, from the cosets of qth power residues and non–residues. These sequences are also shown to fall into two classes, each of which has well–defined periodic autocorrelation functions and merit factors. Class–I sequences exist for prime L ≡ q + 1 mod 2q and q even, whereas class–II sequences correspond to primes of the form L ≡1 mod 2q and are available for all values of q. Class–I sequences have complex out–of–phase correlation values with magnitudes less than or equal to , whereas for class–II sequences these are purely real with magnitude less than or equal to 3. Some other properties are also investigated.