Exercise 8, § 7

Let $k$ be a field and $T$ a finite family of indeterminates. The $k$-algebra $k[[T]]$ is endowed with the discrete topology.

a) Prove that $k[[T]]$ is formally smooth over $k[T]$ relative to $k$ (exerc. 7).

b) If $k[[\mathbf{T}]]$ is formally smooth over $k$, prove that the characteristic $p$ of $k$ is strictly positive and that $k$ is a finite extension of $k^p$ (deduce from exerc. 2 that $\Omega_{k[\mathbf{T}]}(k[[\mathbf{T}]])$ is zero, and consequently that $\Omega_{k(\mathbf{T})}(k((\mathbf{T})))$ is zero; prove then with the aid of prop. 6 of A, V, p. 99, that the subextension of $k((\mathbf{T}))$ formed by the series whose coefficients generate an extension of finite type of $k^p$ is equal to $k((\mathbf{T}))$).