 Every field "K" has an algebraic closure, which is up to an isomorphism the largest extension field of "K" which is algebraic over "K", and also the smallest extension field such that every polynomial with coefficients in "K" has a root in it.For example, formula_6 is an algebraic closure of formula_7 but not an algebraic closure of formula_17 as it is not algebraic over formula_23 (for example is not algebraic over formula_23).Given a field extension , a subset "S" of "L" is called algebraically independent over "K" if no non-trivial polynomial relation with coefficients in "K" exists among the elements of "S".