Field extension

In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties.

Field extensions can be generalized to ring extension which consists of a ring and one of its subrings.

Definitions

Given two fields K and L, if K is a subset of L and the field operations of addition and multiplication in K are the same as those in L, we say that K is a subfield of L, L is an extension field of K and that L/K, read as "L over K", is a field extension.

If L is an extension of F which is in turn an extension of K, then we say F is an intermediate field or subextension of the field extension L/K.

Given a field extension L/K and a subset S of L, we denote by K(S) the smallest subfield of L which contains K and S. We say K(S) is generated by the adjunction of elements of S to K. If S consists of only one element s we often write K(s) instead of K({s}). A field extension of the form L=K(s) is called a simple extension and s is called a primitive element of the extension.

Given a field extension L/K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars". We add the vectors just like we add elements in L, and scalar multiplication is multiplication of elements from L by elements from K. The dimension of this vector space is called the degree of the extension, and is denoted by ^*.

An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.

Notes

The notation L/K is purely formal and does not imply the formation of a quotient ring or quotient group or any other kind of division.

It is often desirable to talk about field extensions in situations where the small field is not actually contained in the larger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every ring homomorphism between fields is injective, so field extensions are precisely the morphisms in the category of fields.

In the sequel, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

Examples

The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q. Clearly then, C/Q is also a field extension. We have = 2 because {1,i} is a basis, so the extension C/R is finite. This is a simple extension because C=R(i). [R : Q = c (the cardinality of the continuum), so this extension is infinite.

The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2 because {1, √2} can serve as a basis. Finite extensions of Q are also called algebraic number fields and are important in number theory.

Another extension field of the rationals, quite different in flavor, is the field of p-adic numbers Q_p for a prime number p.

It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring Kin order to "create" a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x² = -1. Then the polynomial X² + 1 is irreducible in Kideal (X² + 1) generated by this polynomial is maximal, and L = K[X" target="_blank" >^*/(X² + 1) is an extension field of K which does contain an element whose square is −1 (namely the residue class of X).

By iterating the above construction, one can construct the splitting field of any polynomial from K^*. This is an extension field L of K in which the given polynomial splits into a product of linear factors.

If p is any prime number and n is a positive integer, we have a finite field GF(pⁿ) with pⁿ elements; this is an extension field of the finite field GF(p) = Z/pZ with p elements.

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 KThis field of rational functions is an extension field of K. We have [K(X) : K = $\aleph_0$ (aleph-null, the cardinality of countably infinite sets), so this extension is infinite.

Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is an extension field of C, if we identify every complex number with the corresponding constant function defined on M.

Given an algebraic variety V over some field K, then the function field of V, consisting of the rational functions defined on V and denoted by K(V), is an extension field of K.

Elementary properties

If L/K is a field extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of (L,+), and the multiplicative group (K-{0},·) is a subgroup of (L-{0},·). In particular, if x is an element of K, then its additive inverse −x computed in K is the same as the additive inverse of x computed in L; the same is true for multiplicative inverses of non-zero elements of K.

In particular then, the characteristics of L and K are the same.

Algebraic and transcendental elements

If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraic over K. Elements that are not algebraic are called transcendental. As an example:

In C/R, i is algebraic because it is a root of x²+1.
In R/Q, √2 + √3 is algebraic, because it is a root of x⁴-10x²+1
In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (see transcendental number)
In C/R, e is algebraic because it is the root of x-e

The special case of C/Q is especially important, and the names algebraic number and transcendental number are used to describe the complex numbers that are algebraic and transcendental (respectively) over Q.

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it is said to be transcendental. If every element of L except those in K is transcendental over K, then the extension is said to be purely transcendental.

It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular, every finite extension is algebraic. For example,

C/R and Q(√2)/Q, being finite, are algebraic.
R/Q is transcendental, although not purely transcendental.
K(X)/K is purely transcendental.

A simple extension is finite if generated by an algebraic element, and purely transcendental if generated by a transcendental element. So

R/Q is not simple, as it is neither finite nor purely transcendental.

Every field K has an algebraic closure; this is essentially the largest extension field of K that is algebraic over K and it contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure of Q (and also of R).

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. The largest cardinality of an algebraically independent set is called the transcendence degree of L/K. Given any algebraically independent set S over K, then K(S)/K is purely transcendental. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic. Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to the transcendence degree of the extension.

Normal, separable and Galois extensions

A field extension L/K is called normal if every irreducible polynomial in K^* that has a root in L completely factors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension field of F such that L/K is normal and which is minimal with this property.

An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable. A Galois extension is a field extension that is both normal and separable.

A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension L/K, we can consider its automorphism group Aut(L/K), consisting of all field automorphisms α : L → L with α(x) = x for all x in K. In case the extension is Galois, this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are called abelian extensions.

For a given field extension L/K, one is often interested in the intermediate fields F (subfields of L that contain K). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.