- #1
Karl Karlsson
- 104
- 12
I thought i understood the theorem below:
i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field
Then this example came up:
The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix $$J=
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}$$
has ##q(x)## as minimal polynomial. $$K=\{p(J):p(x)\in \mathbb{R}[x]\}=\{aI+bJ:a,b\in \mathbb{R}\}$$
is a finite field that is isomorphic to ##\mathbb{C}##.
Why can't the finite field K above not be for example ##\{aI+bJ^2+cJ^3:a,b,c\in \mathbb{R}\}##, since k[x] in the theorem i) is ##\mathbb{R}[x]## ?
i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field
Then this example came up:
The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix $$J=
\begin{pmatrix}
0 & -1 \\
1 & 0
\end{pmatrix}$$
has ##q(x)## as minimal polynomial. $$K=\{p(J):p(x)\in \mathbb{R}[x]\}=\{aI+bJ:a,b\in \mathbb{R}\}$$
is a finite field that is isomorphic to ##\mathbb{C}##.
Why can't the finite field K above not be for example ##\{aI+bJ^2+cJ^3:a,b,c\in \mathbb{R}\}##, since k[x] in the theorem i) is ##\mathbb{R}[x]## ?