In field theory and number theory, an element of an extension field of is algebraic over if it is a root of some polynomial with coefficients in . The minimal polynomial over of is thus the monic polynomial of minimal degree that has as a root. Because is a field, this minimal polynomial is necessarily irreducible over . For example, the minimal polynomial (over the reals as well as over the rationals) of the complex number is formula_36. The cyclotomic polynomials are the minimal polynomials of the roots of unity.