Consider matrices of the form $\begin{bmatrix} a & b \\ -b & a-b \end{bmatrix}$ with entries from $\mathbb{R}$, closed under addition and matrix multiplication (see Unital rings within matrices ). This forms a unital and commutative ring.
Moreover, the determinant of such a matrix is of the form $x^2 + y^2 - xy = f(x,y)$ for $x,y \in \mathbb{R}$. Notice that $f_x = 2x - y$ and $f_y = 2y - x$, so the only possible extremum is at the critical point $(0,0)$, and for large $(x,y)$ we have positive $f(x,y)$, so I suspect that $f(x,y) > 0$ for $(x,y) \neq (0,0)$. So all nonzero matrices of this form are invertible, which means this ring is a field. Call it $M$.
Is there a more well-known field $K$ such that $M \cong K$? I notice that $M \cong \mathbb{R}^2$ with multiplication defined as $(a,b) * (c,d) = (ac-bd, ad+bc-bd)$, which is close to defining it as $(ac-bd, ad+bc)$ for $\mathbb{C}$.
EDIT: there are much simpler ways to prove $x^2+y^2-xy \neq 0$ if $(x,y) \neq (0,0)$. How can I prove that $xy\leq x^2+y^2$?
