Consider the following polynomials
- $x^2+1$, this doesn't factor over $\mathbb{R}$ but it does factor over $\mathbb{C}$ as $(x+i)(x-i)$
- Consider: $ (x^2 + y^2 + z^2 - k^2)$, this doesn't factor over $\mathbb{R}$ or $\mathbb{C}$ but as many physicists know, over the space of 4x4 complex matrices, its possible to find matrices $A,B,C,D$ such that $(Ax+By+Cz+Dk)^2$ = $(I_4 x^2 + I_4 y^2 + I_4 z^2 - I_4 k^2)$ and so this polynomial can be said to factor over the space of 4x4 complex matrices (There should be a better term for that space of matrices equipped with multiplication and addition but I have forgotten it)
Even then there are polynomials which simply cannot be factored with finite matrices, an example is:
$xy+1$ , In this particular case, what sort of algebraic object generalizing matrices and matrix multiplication the way matrices do to complex numbers, is necessary to factor such an expression into linear sub-factors?
Considerations:
the above can’t be factored in the form $(A+Bx)(C+Dy)$ where $A,B,C,D$ are matrices since the resulting system $AC=BD=1, AD=BC=0$ has no solutions by a determinant argument.
If we look instead for factors of the form $(A+Bx+Cy)(D+Ex+Fy)$ then the systems yield $BF+CE=AD=1, AE+BD=AF+CD=BE=CF=0$ and this doesn’t immediately appear intractable.
So it seems the form of factorization is an important thing to be mindful of.