The Generic Minimal Polynomial

Abstract

Let A be a Jordan algebra over $\mathbb{C}$. We assume that A is finite dimensional (as a vector space) and that A has a unit element e. For x ∈ A and each polynomial $p \in \mathbb{C}[T]$,

$$ p = {a_0} + {a_1}T +...+ {a_n}{T^m} $$

we denote by p(x) the element of A defined by

$$ p(x) = {a_0}e + {a_1}x +...{a_m}{x^m} $$

(4.1)

which is called the value of p at x. As the algebra A is power associative, the map

$$ p \mapsto p(x):\mathbb{C}[T] \to A $$

is an algebra homomorphism. Its image

$$ k{\text{[x] = }}\left\{ {p(x)\left| {p \in \mathbb{C}[T]} \right.} \right\} $$

(4.2)

is an associative subalgebra of A and its kernel

$$ {\mathcal{I}_x} = \{ p \in \mathbb{C}[T]\left| {p(x) = 0} \right.\} $$

(4.3)

is an ideal of $\mathbb{C}{\text{[T]}}$. The monic generator m_x of I_x is the minimal polynomial of x; its degree is the rank of x, denoted by rk(x). So we have

$$ rk(x) = \min \{ k\left| {k > 0,e \wedge x} \right. \wedge... \wedge {x^k} = 0\} $$

(4.4)

Where $e \wedge x \wedge...{x^k}$ is computed in the (k + 1)-th exterior power of the vector space A. The rank of the Jordan algebra A is the maximal rank of elements of A:

$$ rk(A)\max {\text{\{ rk(x)}}\left| {x \in A} \right.{\text{\} }} $$

(4.5)

.