This follows since the minimal polynomial is separable, because the roots of unity are distinct.

We assume (as we obviously can) that is the minimal polynomial of .

The minimal polynomial of this matrix is x + 2.

This is equivalent to the minimal polynomial being square-free.

This is the minimal polynomial of a and it encodes many important properties of a.

Because it has a root of absolute value 1, the minimal polynomial for a Salem number must be reciprocal.

This value is also because the product of the three roots of the minimal polynomial is 1.

The minimal polynomial is often the same as the characteristic polynomial, but not always.

This follows because its minimal polynomial over the rationals has degree 3.

Therefore we can speak about the trace of d, the characteristic and minimal polynomials.