In field theory, a branch of mathematics, a minimal polynomial is defined relative to a field extension E/F and an element of the extension field E. The minimal polynomial of an element, if it exists, is a member of F[x], the  ring of polynomials in the variable x with coefficients in F. Given an element a of E, let J_a be the set of all polynomials f(x) in F[x] such that f(a) = 0. The element a is called a root or zero of each polynomial in J_a. The set J_a is so named because it is an ideal of F[x]. The zero polynomial, whose every coefficient is 0, is in every J_a since 0aⁱ = 0 for all a and i. This makes the zero polynomial useless for classifying different values of a into types, so it is excepted. If there are any non-zero polynomials in J_a, then a is called an algebraic element over F, and there exists a monic polynomial of least degree in J_a. This is the minimal polynomial of a with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of J_a, then a is called a transcendental element over F and has no minimal polynomial with respect to E/F.