Given algebraic numbers a_1, ..., a_n it is always possible to find a single algebraic number b such that each of a_1, ..., a_n can be expressed as a polynomial in b with rational coefficients. The number b is then called a primitive element of the extension field Q(a_1, ..., a_n)/Q. Stated differently, an algebraic number b is a primitive element of Q(a_1, ..., a_n)/Q iff Q(a_1, ..., a_n) = Q(b). Primitive elements were implemented in version of the Wolfram Language prior to 6 as PrimitiveElement[z, { a1, ..., an}] (after loading the package NumberTheoryˋPrimitiveElementˋ.

extension field | primitive polynomial | primitive root