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 *α* of *E*, let *J*_{α} be the set of all polynomials *f*(*x*) in *F*[*x*] such that *f*(*α*) = 0. The element *α* is called a root or zero of each polynomial in *J*_{α}. The set *J*_{α} is so named because it is an ideal of *F*[*x*]. The zero polynomial, whose every coefficient is 0, is in every *J*_{α} since 0*α*^{i} = 0 for all *α* and *i*. This makes the zero polynomial useless for classifying different values of *α* into types, so it is excepted. If there are any non-zero polynomials in *J*_{α}, then *α* is called an algebraic element over *F*, and there exists a monic polynomial of least degree in *J*_{α}. This is the minimal polynomial of *α* with respect to *E*/*F*. It is unique and irreducible over *F*. If the zero polynomial is the only member of *J*_{α}, then *α* is called a transcendental element over *F* and has no minimal polynomial with respect to *E*/*F*.