Show that the ring of all matrices of given form is isomorphic to a product of 3 fields

Question

Show that the subring $M\subset \mathbb Q^{4\times 4}$ consisting of all matrices of the form $$A=\begin{pmatrix}a&d&c&b\\b&a&d&c\\c&b&a&d\\d&c&b&a\end{pmatrix}$$ is isomorphic to the product of $3$ fields.

As a hint the following matrix was given: $$P=\begin{pmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}$$ With this, we can see that $A=aP^4+bP+cP^2+dP^3$ and $P$ is of order $4$. But wouldn't this mean that $\{P,P^2,P^3,P^4\}$ is a $\mathbb Q$-Basis of $M$ and thus $M\cong\mathbb Q\times\mathbb Q\times\mathbb Q\times\mathbb Q$ which makes it a product of $4$ fields? Hopefully someone here can point out my mistake.

Answer 1

Your mistake is that $M\cong\mathbb Q^4$ as $\mathbb Q$-vector spaces, not as rings.

Hint: $M \cong \mathbb Q[x]/(x^4-1)$ as rings.

Solution:

$x^4-1 = (x-1)(x+1)(x^2+1) \implies \mathbb Q[x]/(x^4-1) \cong \mathbb Q \times \mathbb Q \times \mathbb Q[i]$