Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
Equivalently, a subfield is a subset that contains the multiplicative identity
, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of
The dimension of this vector space is called the degree of the extension and is denoted by
This is the primitive element theorem, which does not hold true for fields of non-zero characteristic.
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.
Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of p-adic numbers
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 K[X] in order to "create" a root for a given polynomial f(X).
is irreducible in K[X], consequently the ideal generated by this polynomial is maximal, and
is an extension field of K which does contain an element whose square is −1 (namely the residue class of X).
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, there is a unique (up to isomorphism) finite field
Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by
is algebraic over K if it is a root of a nonzero polynomial with coefficients in K. For example,
In this case the degree of the extension equals the degree of the minimal polynomial, and a basis of the K-vector space K(s) consists of
, 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.
is said to be purely transcendental if and only if there exists a transcendence basis S of
If L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarily follow that L = K(S).
On the opposite, even when one knows a transcendence basis, it may be difficult to decide whether the extension is purely separable, and if it is so, it may be difficult to find a transcendence basis S such that L = K(S).
Purely transcendental extensions of an algebraically closed field occur as function fields of rational varieties.
The problem of finding a rational parametrization of a rational variety is equivalent with the problem of finding a transcendence basis that generates the whole extension.
is called normal if every irreducible polynomial in K[X] 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
A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple).
, consisting of all field automorphisms α: L → L with α(x) = x for all x in K. When the extension is Galois this automorphism group is called the Galois group of the extension.
, one is often interested in the intermediate fields F (subfields of L that contain K).
A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field.
CSAs can be further generalized to Azumaya algebras, where the base field is replaced by a commutative local ring.
Given a field extension, one can "extend scalars" on associated algebraic objects.
In addition to vector spaces, one can perform extension of scalars for associative algebras defined over the field, such as polynomials or group algebras and the associated group representations.