Exercise 1, § 4

Let $u(X) = \sum_{n=0}^m \alpha_n X^n$ be a formal power series over a commutative field $K$. a) For $u$ to be a rational fraction of $K(X)$ it is necessary and sufficient that there should exist a finite sequence of elements $(\lambda_i)_{1 \leq i \leq q}$ of elements of $K$, not all zero, and an integer $d \geq 0$ such that for all $n \geq d$ we have

b) Put (« Hankel determinant ») show that if $H_{d+i}^{(q+1)} = 0$ and $H_{d+i}^{(q)} \neq 0$ for every integer $j \geq 0$, then $u(X)$ is a rational fraction (use a)). c) Prove the identity (cf. III, p. 639, Exercise 10). Deduce that if $H_{m+j}^{(k+1)} = 0$ for $0 \leq j \leq r-1$, then the $r$ determinants $H_{m+j}^{(k)}$ where $1 \leq j \leq r$ are either all zero or all non-zero. d) Deduce from b) and c) that for $u(X)$ to be a rational fraction it is necessary and sufficient that there exist two integers $d$ and $q$ such that $H_{d+i}^{(q+1)} = 0$ for all integers $j \geq 0$. e) Let $k$ be a subfield of $K$, and identify $K(X)$, $k(X)$ and $k[[X]]$ with subrings of $K((X))$. Prove that $k[[X]] \cap K(X) = k[[X]] \cap k(X)$.