Exercise 18, § 14

Let K be a field and $K((X))$ the field of formal power series in one indeterminate over K (IV, p. 38). Show that $\operatorname{tr.} \deg_K K((X)) = \operatorname{Card}(K^N)$. Distinguish two cases: a) If $\operatorname{Card}(K) < \operatorname{Card}(K^N)$, note that $\operatorname{Card}(K((X))) = \operatorname{Card}(K^N)$. b) If $\operatorname{Card}(K) = \operatorname{Card}(K^N)$, let P be the prime subfield of K, S an infinite set of elements of $K((X))$ which is algebraically independent over K and T the set of coefficients of all the formal power series belonging to S. Let L be the relative algebraic closure in

K((X)) of the field K(S) and u an element of L ; there is an equation $g(s_1, ..., s_m, u) = 0$ where $g \neq 0$ is a polynomial in $K[X_1, ..., X_n, X_{m+1}]$ and $s_1, ..., s_m$ elements of S. Let A be the set of coefficients of the polynomial g and $C(u)$ the set of coefficients of the formal power series u ; show that the field P(T)(A U C(u)) is algebraic over P(T U A ) (denoting by $\Omega$ an algebraic closure of K, show that otherwise there would exist an infinity of formal power series $v \in \Omega((X))$ satisfying the equation $g(s_1, ..., s_m, v) = 0$. Using Ex. 1, show that if Card(S) < Card(K), then the transcendence degree of K over P(T) is infinite and deduce that in this case L is distinct from K((X)) (if $(t_n)_{n \geq 0}$ is an infinite sequence of elements of K algebraically independent over P(T), consider the formal power series $\sum_{n=0}^{\infty} t_n X^n$).