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$ k$ -rationality

PROPOSITION 4.13   Let $ A$ be a square matrix over a field $ k$ . Let

$\displaystyle A=S+N
$

be the Jordan-Chevalley decomposition of $ A$ .

Let $ m_A$ be the minimal polynomial of $ A$ . If all of the roots of $ m_A$ are separable over $ k$ , then $ S$ and $ N$ are defined over $ k$ . (That means, they are matrices over $ k$ ).

PROOF.. In view of Corollary , we may assume that $ m_A$ is of the form $ f^e$ for some irreducible polynomial $ f$ and a positive integer $ e$ . By assumption, $ f$ has only simple roots.

$\displaystyle f(X)=\prod_{j=1}^d (X-c_j)
$

Let us define a polynomial $ \chi_s(X) \in \overline{k}[X]$ as follows.

$\displaystyle \chi_s(X)=
\frac
{\prod_{{1\leq j \leq d}\atop {j\neq s}} (X-c_j)}
{\prod_{{1\leq j \leq d}\atop {j\neq s}} (c_s-c_j)}
\qquad (s=1,2,3,\dots,d)
$

These polynomials are designed to satisfy the following property.

\begin{displaymath}
\chi_s(c_j)=
\begin{cases}
1 & \text{ if }j=s\\
0 & \text{ if }j\neq s
\end{cases}\end{displaymath}

Then we further define

$\displaystyle \phi_s^{(e)}(X)=1-(1-\chi_s^e)^e
$

and

$\displaystyle \psi(X)=\sum_{s=1}^{d} c_s \phi_s^{(e)}(X).
$

It is fairly easy to see that

$\displaystyle S=\psi(A)
$

holds.

The function $ \psi$ is symmetric with respect to roots $ \{c_s\}$ and thus $ \psi$ is a polynomial with coefficients in $ k$ . Thus $ S$ (hence also $ N$ ) is defined over $ k$ .

$ \qedsymbol$

The following example shows that the $ k$ -rationality of $ S$ does not necessarily hold when we drop off the assumption on $ A$ .

EXAMPLE 4.14   Let $ A\in M_p(\mathbb{F}_p(x))$ be a matrix of the following form.

$\displaystyle A=
\begin{pmatrix}
0& 1 &\\
& \ddots & \ddots\\
& &\ddots& 1 \\
x& & & 0
\end{pmatrix}$

Then the minimal polynomial of $ A$ is given by $ X^p-x$ . The Jordan-Chevalley decomposition of $ A$ is given by

$\displaystyle A=x^{1/p}+(A-x^{1/p}).
$

Thus the decomposition is not defined over $ \mathbb{F}_p(x)$ .


next up previous
Next: generalities in finite dimensional Up: Jordan-Chevalley decomposition of a Previous: Existence and uniqueness of
2009-03-06