Exercise 9, § 6

a) Let K be a field and $X_1, \ldots, X_n$ indeterminates. Show that for any exponents $v_1 < v_2 < \ldots < v_n$ the determinant $\det(X_i^{v_j})$ is non-zero in $K(X_1, \ldots, X_n)$.

b) Let $s_k$ ($1 \leq k \leq n$) be the elementary symmetric polynomials in the $X_i$ and $r_1, \ldots, r_n$ n elements of the field $K(X_1, \ldots, X_n)$. We put (with a new indeterminate T)

Show that if $n$ of the coefficients $u_k$ are zero, then all the $u_k$ are zero. (Decompose $R(T)$ into simple fractions.)

*c) Suppose that K is of characteristic zero. Let $\{v_1, v_2, \ldots, v_n\}$ be a set of n elements $\geq 1$ of $\mathbf{N}$, and let M be the complement of this set in $\mathbf{N}$; suppose also that M is stable under addition. Show that if $y_1, \ldots, y_n$ are n elements of $K(X_1, \ldots, X_n)$ such that

for $1 \leq j \leq n$, then $y_j = X_j$ for $1 \leq j \leq n$ except possibly for a permutation. (Consider the expansion of $\log R(T) = v_1 T + v_2 T^2 + \cdots + v_k T^k + \cdots$, observe that $v_{v_j} = 0$ for $1 \leq j \leq n$ and examine the relations between the $v_k$ and the $u_k$.)