An analogue of the theorem on the existence of a primitive element for separable extensions of fields is presented for semigroups. This has two immediate consequences.
(i) A semigroup is algebraically closed with respect to equations in several variables if and only if it is closed with respect to equations in a single variable.
(ii) Any countable semigroup C is embedded in a two-generator semigroup, one of whose generators is in C.
Further, a proof is given that any free product of a semigroup of order one with one of order two is SQ–universal, that is, its factor semigroups embed all countable semigroups. The proofs are adaptations of one used by Trevor Evans, Proc. Amer. Math. Soc. 3 (1952), 614–620. to show that a free product of two infinite cyclic semigroups is SQ–universal.