Exercise 22, § 8

Let F be a mapping of $M_n(K)$ into a set E, where K is a commutative field, such that, for every triple of matrices $X, Y, Z$ in $M_n(K)$,

We except the case $M_2(F_2)$; show then that there exists a mapping $\Phi$ of K into E such that $F(X) = \Phi(\det(X))$. (Using the Corollary to Proposition 17 (no. 9), show first that, for $U \in SL_n(K)$, $F(X) = F(XU)$; deduce that, in $GL_n(K)$, $F(X)$ depends only on $\det(X)$. On the other hand, if, for $r \leq n - 1$, we write

show that there exists a matrix $Y$ of rank $r$ such that $X_{r-1} = YX_r$ and $Y = X_rY$ and deduce that $F(X)$ has the same value for all matrices of rank $< n$.)