Field. Number field. Extension field. Splitting field. Separable polynomial. Algebraic number. Algebraic element, Transcendental element.

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Field. A set of elements a, b, ... for which two operations, called addition and multiplication, are defined and have the properties:

1. The elements form an Abelian group with addition as the group operation.

2. The elements, with the identity element (0) of the additive group omitted, form an Abelian group with multiplication as the group operation.

3. a(b + c) = ab + ac for all a, b, and c in the set.

Syn. Domain

Number field. Any set of real or complex numbers such that the sum, difference, product, and quotient (except by zero) of any two members of the set is in the set.

Syn. Number domain

Examples.

1. The set of all rational numbers.

2. The set of all numbers of the form

Extension field. In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; 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.

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 trivial extension. 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. The degree of a finite extension L / K is denoted [L : K]

Extension of a field. Any field F* that contains a field F is an extension of F. The degree of the extension is the dimension of F* as a vector space with scalars in F. A finite extension is an extension whose degree is finite. An algebraic extension of F is an extension whose members all satisfy polynomial equations with coefficients in F.

Splitting field. In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

Separable polynomial. A polynomial that has no multiple zeros. A polynomial with coefficients in a field F is separable if and only if the greatest common divisor of f and its formal derivative fʹis a constant.

Separable extension of a field. Let F* be a field that contains a field F. Then a member c of F* is separable with respect to F if c is a zero of a separable polynomial with coefficients in F. The extension F* is separable if all members of F* are separable.

Algebraic number. Any number which is a root of a polynomial equation with rational coefficients. The degree of the algebraic number is the degree of the monic polynomial of least degree that has the number as a root. An algebraic integer is an algebraic number which satisfies some monic equation

xⁿ + a₁ x^n-1+ .... + a_n = 0

with integers as coefficients. A rational number is an algebraic integer if and only if it is an ordinary integer.

Rational number. A number that can be expressed as an integer or a quotient of integers.

Irrational number. A real number not expressible as an integer or quotient of integers; a nonrational number. The irrational numbers are of two types: 1) algebraic irrational numbers (irrational numbers which are the roots of polynomial equations with rational coefficients) and 2) transcendental numbers.

Examples of algebraic numbers:

Examples of transcendental numbers: π, e

Algebraic element, Transcendental element. Let K be any field and F any subfield of K. An element c of K is called algebraic over F if c satisfies a polynomial equation with coefficients not all zero in F,

a₀ + a₁c + a₂c² + .... + a_ncⁿ= 0 (a_i in F, not all 0)

with coefficients not all zero in F. An element of K which is not algebraic over F is called transcendental over F.

Degree of an algebraic element. The degree n of a element u algebraic over a field F is the degree n of the monic irreducible polynomial with coefficients in F and root u.

Example. The monic irreducible polynomial of over the field R of rationals is x⁵ -3 = 0. Thus the degree of is 5 over the rationals.

References

James and James. Mathematics Dictionary.