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Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; the elements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomial ring K[X]. This field of rational functions is an extension field of K. This extension is infinite.
The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.
A field extension L/K is called a simple extension if there exists an element θ in L with = (). This means that every element of L can be expressed as a rational fraction in θ, with coefficients in K; that is, it is produced from θ and elements of K by the field operations +, −, •, / .
In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors in L. [1] [2] This is one of the conditions for an algebraic extension to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.
In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension K / K of the algebraic closure, leading to the absolute Galois group G := Gal(K / K) or just Gal(K), and to the extension /.
Every finite extension of k is separable. Every algebraic extension of k is separable. Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power. Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x ↦ x p is an automorphism of k. The separable closure of k ...
An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom of choice. [11] An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
Radical extensions occur naturally when solving polynomial equations in radicals.In fact a solution in radicals is the expression of the solution as an element of a radical series: a polynomial f over a field K is said to be solvable by radicals if there is a splitting field of f over K contained in a radical extension of K.