- #1

- 91

- 10

Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros.

In the picture below is the given solution for this:

I understand that the eigenvalues must be conjugates. But I don't understand how they so quickly arrived at the finite field extension K

The only theorem I know related to solving the problem above is:

Theorem: 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

Thanks in advance!