［Solution］ Let $x$ be a common root, then we have
$$\left\{\begin{array}{l} x^{2} + a x + 1 = 0 \\ x^{2} + x + a = 0 \end{array}\right.$$
(1) - (2) gives $\square$
$$(a-1)(x-1)=0 .$$

If $a=1$, then equations (1) and (2) are the same, in which case the two polynomials have a common root.
If $a \neq 1$, then the common root $x=1$, substituting into (1), (2) both yield $a=-2$.
Therefore, when $a=1$ or -2, $x^{2} + a x + 1$ and $x^{2} + x + a$ have at least one common root.