I'm not sure if the fundamental theorem of algebra extends to every possible and imaginable numbers (real, complex, quaternions, etc.) but here's my question anyway.
Let $f(x) = x^2-2ax+(a^2+b^2+c^2+d^2)$ and let the quaternion $q=a+bi+cj+dk$, where the coefficients of $q$ are real. We can easily verify that $q$ and $\bar{q}$ are roots of $f(x)$.
But there are 2 other real roots $x_1$ and $x_2$ to $f(x)$:
$$x_1 = a - \sqrt{-b^2-c^2-d^2}$$ $$x_2 = a + \sqrt{-b^2-c^2-d^2} $$
Shouldn't the fundamental theorem of Algebra forbid that? Because I thought that a polynomial of degree $n$ can only have $n$ distinct roots maximum, which is not the case here since $f(x)$ is a 2nd degree polynomial but has 4 root distinct roots.
Can anyone enlighten me, please?
Thank you
(Port of @Randall's comment into an answer)
This is because the Fundamental Theorem of Algebra requires commutativity, and quaternions lack this property.