Exercise 17, § 3

a) Let A be an integral ring; now prove 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 to the case where A is a field; observe that the factors of $\det(X_{ij})$ would necessarily be homogeneous polynomials, and reason by induction on n).

b) Let K be an infinite field, F a polynomial of $K[Y_1, \ldots, Y_m]$, which is also written $F(Y)$; for every square matrix $s = (\alpha_{ij})$ of order m with elements in K, denote by $F(s . Y)$ the polynomial F in which one has substituted for each $Y_i$ the element $\sum_{j=1}^m \alpha_{ij} Y_j$. Now prove that if F is extremal, the same is true of $F(s . Y)$ for every invertible matrix s. If there exists an integer $k \geq 0$ such that $F(s . Y) = (\det(s))^k F(Y)$ for every invertible matrix s, F is necessarily homogeneous with respect to each of the $Y_j$; moreover, if $F = GH$ where G and H are two polynomials of $K[Y_1, \ldots, Y_m]$, there exist two integers $p$ and $q$ such that $p + q = k$ and $G(s.Y) = (\det(s))^p G(Y)$, $H(s.Y) = (\det(s))^q H(Y)$ for every invertible matrix s (use $a$).

c) Let $A$ be a nonzero ring; one considers 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 $i > j$). Now prove that $\det(U)$ (resp. $\mathrm{Pf}(V)$) is an extremal element in $A[X_{ij}]$ (reasoning as in $b$), by considering $\det(s.U.'s)$ and $\mathrm{Pf}(s.V.'s)$.