Algebraic element
In mathematics, if is a field extension of , then an element of is called an algebraic element over , or just algebraic over , if there exists some non-zero polynomial with coefficients in such that . Elements of which are not algebraic over are called transcendental over . These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is , being the field of complex numbers and being the field of rational numbers). Properties. The following conditions are equivalent for an element formula_1 of formula_2: To make this more explicit, consider the polynomial evaluation formula_21. This is a  homomorphism and its kernel is formula_22. If formula_1 is algebraic, this ideal contains non-zero polynomials, but as formula_24 is a euclidean domain, it contains a unique polynomial formula_25 with minimal degree and leading coefficient formula_26, which then also generates the ideal and must be irreducible. The polynomial formula_25 is called the minimal polynomial of formula_1 and it encodes many important properties of formula_1. Hence the ring isomorphism formula_30 obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that formula_31. Otherwise, formula_32 is injective and hence we obtain a field isomorphism formula_33, where formula_34  is the field of fractions of formula_24, i.e. the field of rational functions on formula_4, by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism formula_37 or formula_38. Investigating this construction yields the desired results. This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over formula_4 are again algebraic over formula_4. For if formula_1 and formula_42 are both algebraic, then formula_43 is finite. As it contains the aforementioned combinations of formula_1 and formula_42, adjoining one of them to formula_4 also yields a  finite extension, and therefore these elements are algebraic as well. Thus set of all elements of formula_2 which are algebraic over formula_4 is a field that sits in between formula_2 and formula_4. Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed. The field of complex numbers is an example. If formula_2 is algebraically closed, then the field of algebraic elements of formula_2 over formula_4 is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above. An example for this is the field of algebraic numbers