Field extension

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 that contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution to the equation x² = 2.

Definitions

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under taking inverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L is then said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L over K") is a field extension to signify that L is an extension field of K.

If L is an extension of F which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of the field extension L / K.

Given a field extension L / K and a subset S of L, the smallest subfield of L which contains K and S is denoted by K(S)—i.e. K(S) is the field generated by adjoining the elements of S to K. If S consists of only one element s, K(s) is a shorthand for 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, the larger field L can be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension and is denoted by [L : K].

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.

Caveats

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. Instead the slash expresses the word "over". In some literature the notation L:K is used.

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 non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.

Henceforth, 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 [C : R] = 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] = ${\mathfrak {c}}$ (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. Q(√2, √3) = Q(√2)( √3)={a + b√3 | a, b ∈ Q(√2)}={a + b√2+ c√3+ d√6 | a, b,c,d ∈ Q} is an extension field of both Q(√2) and Q, of degree 2 and 4 respectively. 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 K[X] in 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 K[X], consequently the ideal (X² + 1) generated by this polynomial is maximal, and L = K[X]/(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 a splitting field of any polynomial from K[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, 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 K[X]. This field of rational functions is an extension field of K. 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 a transcendental extension field of C, if we identify every complex number with the corresponding constant function defined on M. More generally, 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.

If we take a non-constant morphism of curves $X\to Y$ then there is an induced finite field extension $K(Y)\to K(X)$ of their function fields. For example, let

${\text{Spec}}\left({\frac {\mathbb {C} [x,y]}{(y^{2}z-x(x-1)(x-2))}}\right)\to {\text{Spec}}(\mathbb {C} [x])$

be a morphism of plane curves. This induces a morphism of integral domains

$\mathbb {C} [X]\to {\frac {\mathbb {C} [X,Y]}{(Y^{2}-X(X-1)(X-2))}}$

inducing a morphism on their associated fraction fields

$\mathbb {C} (X)\to \mathbb {C} (X,Y)/(Y^{2}-X(X-1)(X-2))$ .

Notice that the fraction field of the elliptic curve is a rank 2 vector space over $\mathbb {C} (X)$ , equal to $\mathbb {C} (X)\oplus \mathbb {C} (X)\cdot y$ . These types of example can be generalized further by looking at morphisms of projective curves, such as a ramified galois covers of $\mathbb {P} ^{1}$

${\text{Proj}}\left({\frac {\mathbb {C} [x,y,z]}{(x^{n}+y^{n}+z^{n})}}\right)\to {\text{Proj}}\left(\mathbb {C} [x,y]\right)$ ,

and looking at an affine chart. Since the function fields of curves are a birational invariant, the affine curves give the function fields of these projective curves.

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 and extensions

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. For 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^[1] 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 a transcendental extension.

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. 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. An extension L/K is said to be purely transcendental if and only if there exists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of L except those of K are transcendental over K, but, however, there are extensions with this property which are not purely transcendental—a class of such extensions take the form L/K where both L and K are algebraically closed. In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarily follow that L=K(S). (For example, consider the extension Q(x,√x)/Q, where x is transcendental over Q. The set {x} is algebraically independent since x is transcendental. Obviously, the extension Q(x,√x)/Q(x) is algebraic, hence {x} is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for √x. But it is easy to see that {√x} is a transcendence basis that generates Q(x,√x)), so this extension is indeed 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 which is algebraic over K and which contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure of R.

Normal, separable and Galois extensions

An algebraic extension L/K 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 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, i.e., has no repeated roots in an algebraic closure over K. 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. When 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.

Generalizations

Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. 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. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be further generalized to Azumaya algebras, where the base field is replaced by a commutative local ring.

Extension of scalars

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vector space, one can produce a complex vector space via complexification. 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. Extension of scalars of polynomials is often used implicitly, by just considering the coefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars has numerous applications, as discussed in extension of scalars: applications.

Notes

↑ "Wolfram|Alpha input: sqrt(2)+sqrt(3)". Retrieved 2010-06-14.

References

Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Extension of a field", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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