Exercise 12, § 2

Let K be a commutative field, A the polynomial ring $K[X_i]_{i \in I}$ in an arbitrary infinite family of indeterminates and m the (maximal) ideal of A generated by the $X_i$. If we set $A_i = A/m^{i+1}$, the $A_i$, and the canonical homomorphisms $h_{ij} : A/m^{j+1} \to A/m^{i+1}$ for $i < j$ satisfy the conditions of Proposition 14; the ring $\lim \leftarrow A$, is the completion $\hat{A}$ of $A$ with respect to the m-adic topology and the kernel of the canonical homomorphism $\hat{A} \to A$, is equal to $(m^t)^*$, the closure of $m^t$ in $A$.

(a) Show that $\hat{A}$ is canonically identified with the ring of formal power series in the $X_n$, each of which has only a finite number of terms of given degree (Algebra, Chapter IV, § 5, Exercise 1).

(b) From now on take $I = \mathbf{N}$. Show that $m \neq \hat{A}.m$ (consider the formal power series $\sum_{n=1}^\infty X_n^n$).

(c) Suppose that $K$ is a finite field. Show that $(\hat{m})^2 \neq (m^2)^*$. (Show first the following result: for every integer $k > 0$, there exists an integer $n_k$ and for every integer $n \geq n_k$ a homogeneous polynomial $F$, of degree $n$ in $n^2$ indeterminates with coefficients in $K$ such that $F$, cannot be the sum of the terms of degree $n$ in any polynomial of the form $P_1 Q_1 + \ldots + P_k Q_k$, where the $P_i$ and $Q_i$ are polynomials without constant term in the same $n^2$ indeterminates).

Deduce that $\hat{A}$ is not complete with respect to the m-adic topology.