Remarks

Note that a polynomial of C[X] lies in Q[X] if and only if it has rational coefficients.

An algebraic number is characterised by the fact that it generates a subfield of C that is finite-dimensional, when viewed as a vector space over Q. For instance, e and are known not to be algebraic (although the proof is not easy).

If a is algebraic, then there is a polynomial of minimal degree of which a is a zero. For, if f and g are both nonzero polynomials of which a is a zero, then so is gcd(f,g).

The notion of algebraic element exists for any field K with a subfield L: an element of K is called algebraic over L if it is a zero of a nonzero polynomial in K[X].