Can Galois theory be applied to solving polynomial matrix equations of a given degree $n$?

Question

Suppose we define a structure $\mathbb{C}[X]$ such that all elements are polynomials $$p_{n}(X)=\sum_{1\le i\le n}C_i X^i$$ where $C_i$ and $X$ are, respectively, coefficient matrices and an unknown matrix $X$ [we are positing that these are square $m$ by $m$ matrices across the board].

My question is almost a pointless question, but humor me: Do we arrive at the same results as in the case of regular polynomials when we perform algebraic reasoning on these objects using Galois theory? Does Galois theory still hold up for being applied to these matrix polynomials?

More specifically: My intuition tells me that there exists a matrix formula for each regular formula relative to degrees 1, 2, 3, and 4, but not 5 or higher (the Abel-Ruffini theorem). It seems to me that this should be a natural result: matrices operate just like scalars in terms of polynomial equations. However, the generality of it is what has me worried because it takes only one circumstance/counterexample to render an entire general theorem such as that incorrect.

Answer 1

Matrix polynomials usually have uncountably many roots (over $\mathbb{R}$ or $\mathbb{C}$). As a simple example, the equation $X^2 = 1$ has solutions given by any diagonalizable matrix with eigenvalues $\pm 1$. More generally if $f(x) = \sum f_i x^i$ is a polynomial with scalar coefficients then we can find matrices satisfying $f(X) = 0$ by finding diagonalizable matrices whose eigenvalues are roots of $f(x)$ in the usual sense and there are usually uncountably many of these.

I don't see how to get anything that looks like Galois theory out of this. Things get even worse once you consider matrices as coefficients; there I'm not aware of any general theory at all. Already the question of characterizing the solutions to $X^2 = C$ for $C$ a matrix is delicate; see this math.SE question and its answers.

It seems to me that this should be a natural result: matrices operate just like scalars in terms of polynomial equations.

They certainly do not; most basically, matrices don't necessarily commute.