Thread: algebraic numbers View Single Post
 2020-03-04, 14:17 #7 Nick     Dec 2012 The Netherlands 165010 Posts 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 non-negative integer n such that $$1,z,z^2,\ldots,z^{n-1}$$ 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 $$1,z,z^2,\ldots,z^{n-1}$$ and therefore z is the root of a monic polynomial of degree n.