information:filtering, structural:summarization, stylistic:tone, structural:expansion
Algebraic extensions

A pivotal notion in the study of field extensions F / E are algebraic elements. An element x ∈ F is algebraic over E if it is a root of a polynomial with coefficients in E, that is, if it satisfies a polynomial equation

with e,..., e in E, and e ≠ 0.For example, the imaginary unit i in C is algebraic over R, and even over Q, since it satisfies the equation

A field extension in which every element of F is algebraic over E is called an algebraic extension. Any finite extension is necessarily algebraic, as can be deduced from the above multiplicativity formula.

The subfield E (x) generated by an element x, as above, is an algebraic extension of E if and only if x is an algebraic element. That is to say, if x is algebraic, all other elements of E (x) are necessarily algebraic as well. Moreover, the degree of the extension E (x) / E, i.e., the dimension of E (x) as an E -vector space, equals the minimal degree n such that there is a polynomial equation involving x, as above. If this degree is n, then the elements of E (x) have the form

For example, the field Q (i) of Gaussian rationals is the subfield of C consisting of all numbers of the form a + bi where both a and b are rational numbers: summands of the form i (and similarly for higher exponents) do not have to be considered here, since a + bi + ci can be simplified to a − c + bi.