Exercise 17, § 3

(a) Let A be an integral domain; show that in the ring $A[X_{ij}]$, where $(X_{ij})$ is a family of $n^2$ indeterminates $(1 \leq i \leq n, 1 \leq j \leq n)$, the element $\det(X_{ij})$ is extremal. (Reduce it to the case where A is a field; observe that the factors of $\det(X_{ij})$ would necessarily be homogeneous polynomials and argue by induction on n) (b) Let K be an infinite field and F a polynomial in $K[Y_1, \ldots, Y_m]$, which is also written as $F(Y)$; for every square matrix $s = (\alpha_{ij})$ of order m with elements in K let $F(s, Y)$ denote the polynomial F, where the element $\sum_{j=1}^m \alpha_{ij} Y_j$ has been substituted for each $Y_j$. Show that, if F is extremal, so is $F(s, Y)$ for every invertible matrix s. If there exists an integer $k \geq 0$ such that

for every invertible matrix s, F is necessarily homogeneous in each of the $Y_j$;

moreover, if $F = GH$ where G and H are two polynomials in $K[Y_1, \ldots, Y_m]$, there exist two integers $p$ and $q$ such that $p + q = k$ and

for every invertible matrix $s$ (use (a)).

(c) Let A be a ring which is not reduced to 0; consider the polynomial ring $A[X_{ij}]$, where the $X_{ij}$ are $n(n+1)/2$ (resp. $2n(2n-1)/2$) indeterminates with $1 \leq i \leq j \leq n$ (resp. $1 \leq i < j \leq 2n$); let $U = (\xi_{ij})$ (resp. $V = (\eta_{ij})$) be the square matrix of order n (resp. $2n$) over $A[X_{ij}]$ such that $\xi_{ij} = X_{ij}$ for $1 \leq i \leq j \leq n$ and $\xi_{ij} = X_{ji}$ for $i > j$ (resp. $\eta_{ii} = 0$ for $1 \leq i \leq 2n$, $\eta_{ij} = X_{ij}$ for $1 \leq i < j \leq 2n$, $\eta_{ij} = -X_{ji}$ for $i > j$). Show that $\det(U)$ (resp. $\mathrm{Pf}(V)$) is an extremal element in $A[X_{ij}]$ (argue as in (b), considering $\det(s \cdot U \cdot t_s)$ and $\mathrm{Pf}(s \cdot V \cdot t_s)$).