Exercise 22, § 2

If $K$ is a commutative field, show that in the ring of formal power series $K[[X_1, X_2, \ldots, X_p]]$ there is only a single maximal ideal, equal to the set of non-invertible elements. On the other hand, show that in the polynomial ring $K[X_1, \ldots, X_p]$ for $p \geq 1$ there always exist several distinct maximal ideals.