If you're comfortable with linear algebra, that is probably the easiest way to understand it.
The complex numbers form a vector space over the rational numbers.
Take any complex number z and consider the sequence:
\[ 1,z,z^2,z^3,\ldots\]
It can happen that the terms remain linearly independent however far we go.
In that case we call z a transendental number.
Otherwise there exists a nonnegative integer n such that
\(1,z,z^2,\ldots,z^{n1}\) are linearly independent but
\(1,z,z^2,\ldots,z^n\) are not.
This is the case in which we call z an algebraic number.
It then follows that \(z^n\) can be written as a linear combination of \(1,z,z^2,\ldots,z^{n1}\)
and therefore z is the root of a monic polynomial of degree n.
