Exercise 2, § 17

Let B be an algebra of finite degree $n$ over the commutative field K. For any element $b$ of B, denote by Pm$_{B/K}(b; X)$ the minimal polynomial of $b$ over K (V, §3, No. 1, p. 16). We have Pm$_{B/K}(b; X) =$ Pm$_{B^o/K}(b; X)$.

a) Prove that Pm$_{B/K}(b; X)$ divides Pc$_{B/K}(b; X)$, which divides (Pm$_{B/K}(b; X)$)$^n$.

b) For every extension L of K, we have Pm$_{B_{(L)}/L}(1\otimes b; X) =$ Pm$_{B/K}(b; X)$.

c) Let $(e_i)_{i\in I}$ be a basis of B over K and $\mathbf{Y}= (Y_i)_{i\in I}$ a family of variables. The principal polynomial of the algebra B with respect to the basis $(e_i)$, denoted by $P((e_i); X;\mathbf{Y})$ or simply $P(X;\mathbf{Y})$, is the minimal polynomial over $K(\mathbf{Y})$ of the element $\sum_{i\in I}Y_ie_i$ of $B_{(K(\mathbf{Y}))}.*$Prove that it belongs to $K[X;\mathbf{Y}]$ (cf. AC, VII, §3, n$^o5$, p. 230, lemme 1 and théorème 2).

d) Let $(e'_i)$ be another basis of B over K and $(\lambda_{ij})$ the matrix with entries in K such that $e_i$ = $\sum_j\lambda_{ij}e'_j$ for $i\in$ I; if we set $Y'_i$ = $\sum_j\lambda_{ji}Y_j$, then $P((e_i); X;\mathbf{Y}) = P((e'_i); X;\mathbf{Y}')$. Let $b=\sum_i\beta_ie_i$ be an element of B; the polynomial $P((e_i); X;\beta_1, . . . , \beta_n)$ is independent of the choice of the basis $(e_i)$; we call it the principal polynomial of $b$ over K. It divides Pc$_{B/K}(b; X)$ and is a multiple of Pm$_{B/K}(b; X).*$

e) Let $B'$ be another K-algebra of finite degree; calculate the principal polynomial of the algebra $B\times B'$ for a suitable basis.