Field extension

In mathematics, particularly in algebra, a field extension is a pair of fields $K\; \backslash subseteq\; L$, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the are an extension field of the ; the real numbers are a subfield of the complex numbers.

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.

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Subfield A subfield $K$ of a field $L$ is a subset $K\backslash subseteq\; L$ that is a field with respect to the field operations inherited from $L$. Equivalently, a subfield is a subset that contains the multiplicative identity $1$, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of $K$.

As , the latter definition implies $K$ and $L$ have the same zero element.

For example, the field of is a subfield of the , which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphism to) a subfield of any field of characteristic $0$.

The characteristic of a subfield is the same as the characteristic of the larger field.

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Extension field If $K$ is a subfield of $L$, then $L$ is an extension field or simply extension of $K$, and this pair of fields is a field extension. Such a field extension is denoted $L/K$ (read as "$L$ over $K$").

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 $L/K$.

Given a field extension $L/K$, the larger field $L$ is a $K$-vector space. The dimension of this vector space is called the degree of the extension and is denoted by $L:K$.

The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a . Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree.

Given two extensions $L/K$ and $M/L$, the extension $M/K$ is finite if and only if both $L/K$ and $M/L$ are finite. In this case, one has

$M=M\backslash cdotL.$

Given a field extension $L/K$ and a subset $S$ of $L$, there is a smallest subfield of $L$ that contains $K$ and $S$. It is the intersection of all subfields of $L$ that contain $K$ and $S$, and is denoted by $K(S)$ (read as "$K$ $S$"). One says that $K(S)$ is the field generated by $S$ over $K$, and that $S$ is a generating set of $K(S)$ over $K$. When $S=\backslash \{x\_1,\; \backslash ldots,\; x\_n\backslash \}$ is finite, one writes $K(x\_1,\; \backslash ldots,\; x\_n)$ instead of $K(\backslash \{x\_1,\; \backslash ldots,\; x\_n\backslash \}),$ and one says that $K(S)$ is over $K$. If $S$ consists of a single element $s$, the extension $K(s)/K$ is called a simple extension and $s$ is called a primitive element of the extension.

An extension field of the form $K(S)$ is often said to result from the of $S$ to $K$.

In characteristic 0, every finite extension is a simple extension. This is the primitive element theorem, which does not hold true for fields of non-zero characteristic.

If a simple extension $K(s)/K$ is not finite, the field $K(s)$ is isomorphic to the field of rational fractions in $s$ over $K$.

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

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

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

The field

$\backslash Q(\backslash sqrt\{2\})\; =\; \backslash left\; \backslash \{\; a\; +\; b\backslash sqrt\{2\}\; \backslash mid\; a,b\; \backslash in\; \backslash Q\; \backslash right\; \backslash \},$

is an extension field of $\backslash Q,$ also clearly a simple extension. The degree is 2 because $\backslash left\backslash \{1,\; \backslash sqrt\{2\}\backslash right\backslash \}$ can serve as a basis.

The field

$\backslash begin\{align\}$

\Q\left(\sqrt{2}, \sqrt{3}\right) &= \Q \left(\sqrt{2}\right) \left(\sqrt{3}\right) \\ &= \left\{ a+b\sqrt{3} \mid a,b \in \Q\left(\sqrt{2}\right) \right\} \\ &= \left\{ a + b \sqrt{2} + c\sqrt{3} + d\sqrt{6} \mid a,b,c, d \in \Q \right\}, \end{align}

is an extension field of both $\backslash Q(\backslash sqrt\{2\})$ and $\backslash Q,$ of degree 2 and 4 respectively. It is also a simple extension, as one can show that

$\backslash begin\{align\}$

\Q(\sqrt{2}, \sqrt{3}) &= \Q (\sqrt{2} + \sqrt{3}) \\ &= \left \{ a + b (\sqrt{2} + \sqrt{3}) + c (\sqrt{2} + \sqrt{3})^2 + d(\sqrt{2} + \sqrt{3})^3 \mid a,b,c, d \in \Q\right\}. \end{align}

Finite extensions of $\backslash Q$ are also called algebraic number fields and are important in number theory. Another extension field of the rationals, which is also important in number theory, although not a finite extension, is the field of $\backslash Q\_p$ 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 $X^2+1$ is irreducible in KX, consequently the ideal generated by this polynomial is maximal ideal, and $L\; =\; KX/(X^2+1)$ 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, there is a unique (up to isomorphism) finite field $GF(p^n)\; =\; \backslash mathbb\{F\}\_\{p^n\}$ with pⁿ elements; this is an extension field of the prime field $\backslash operatorname\{GF\}(p)\; =\; \backslash mathbb\{F\}\_p\; =\; \backslash Z/p\backslash Z$ 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 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 $\backslash Complex(M).$ It is a transcendental extension field of $\backslash Complex$ 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, the function field K( V), consisting of the rational functions defined on V, is an extension field of K.

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Algebraic extension An element x of a field extension $L/K$ is algebraic over K if it is a root of a nonzero polynomial with coefficients in K. For example, $\backslash sqrt\; 2$ is algebraic over the rational numbers, because it is a root of $x^2-2.$ If an element x of L is algebraic over K, the monic polynomial of lowest degree that has x as a root is called the minimal polynomial of x. This minimal polynomial is irreducible over K.

An element s of L is algebraic over K if and only if the simple extension is a finite extension. 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 $1,\; s,\; s^2,\; \backslash ldots,\; s^\{d-1\},$ where d is the degree of the minimal polynomial.

The set of the elements of L that are algebraic over K form a subextension, which is called the algebraic closure of K in L. This results from the preceding characterization: if s and t are algebraic, the extensions and are finite. Thus is also finite, as well as the sub extensions , and (if ). It follows that , st and 1/ s are all algebraic.

An algebraic extension $L/K$ is an extension such that every element of L is algebraic over K. Equivalently, an algebraic extension is an extension that is generated by algebraic elements. For example, $\backslash Q(\backslash sqrt\; 2,\; \backslash sqrt\; 3)$ is an algebraic extension of $\backslash Q$, because $\backslash sqrt\; 2$ and $\backslash sqrt\; 3$ are algebraic over $\backslash Q.$

A simple extension is algebraic if and only if it is finite. This implies that an extension is algebraic if and only if it is the union of its finite subextensions, and that every finite extension is algebraic.

Every field K has an algebraic closure, which is up to an isomorphism the largest extension field of K which is algebraic over K, and also the smallest extension field such that every polynomial with coefficients in K has a root in it. For example, $\backslash Complex$ is an algebraic closure of $\backslash R$, but not an algebraic closure of $\backslash Q$, as it is not algebraic over $\backslash Q$ (for example is not algebraic over $\backslash Q$).

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Transcendental extension Given a field extension $L/K$, 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 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.

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).

For example, consider the extension $\backslash Q(x,\; y)/\backslash Q,$ where $x$ is transcendental over $\backslash Q,$ and $y$ is a polynomial root of the equation $y^2-x^3=0.$ Such an extension can be defined as $\backslash Q(X)Y/\backslash langle\; Y^2-X^3\backslash rangle,$ in which $x$ and $y$ are the equivalence classes of $X$ and $Y.$ Obviously, the singleton set $\backslash \{x\backslash \}$ is transcendental over $\backslash Q$ and the extension $\backslash Q(x,\; y)/\backslash Q(x)$ is algebraic; hence $\backslash \{x\backslash \}$ is a transcendence basis that does not generate the extension $\backslash Q(x,\; y)/\backslash Q(x)$. Similarly, $\backslash \{y\backslash \}$ is a transcendence basis that does not generates the whole extension. However the extension is purely transcendental since, if one set $t=y/x,$ one has $x=t^2$ and $y=t^3,$ and thus $t$ generates the whole extension.

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.

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Normal, separable and Galois extensions An algebraic extension $L/K$ is called normal extension 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 $\backslash text\{Aut\}(L/K)$, consisting of all field 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 group 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 of the Galois group, described by the fundamental theorem of Galois theory.

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Generalizations Field extensions can be generalized to ring extensions which consist of a ring and one of its . 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 , where the base field is replaced by a commutative local ring.

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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 ring 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 .

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See also

• Field theory
• Glossary of field theory
• Tower of fields
• Primary extension
• Regular extension

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Notes

• edition=3r2004

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External links

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