In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...

X − a_i,

and such that the a_i generate L over K. It can be shown that such splitting fields exist, and are unique up to isomorphism; the amount of freedom in that isomorphism is known to be the Galois group of P (if we assume it is separable, anyway). Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, a polynomial P(X) is separable over a field K if its roots in an algebraic closure of K are distinct - that is P(X) has distinct linear factors in some large enough field extension. ...

For an example if K is the rational number field Q and In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

P(X) = X³ − 2,

then a splitting field L will contain a cube root of unity, as well as a cube root of 2. In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ...

Given an algebraically closed field A containing K, there is a unique splitting field L of P between K and A, generated by the roots of P. In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ...

Therefore, for example, for K given as a subfield of the complex numbers, the existence is automatic. On the other hand the existence of algebraic closures in general is usually proved by 'passing to the limit' from the splitting field result; which is therefore proved directly to avoid a vicious circle. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Vicious Circle is an album released in 1995 by L.A. Guns. ...

Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials P over K that are minimal polynomials over K of elements a of K′. In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ...

Examples

• The splitting field of x² + 1 over R, the real numbers, is C, the complex numbers.
• The splitting field of x² + 1 over GF₇ is GF_7².
• The splitting field of x² − 1 over GF₇ is GF₇ since x² − 1 = (x + 1)(x − 1) already factors into linear factors.

See also

• Construction of splitting fields

In mathematics, a splitting field of a polynomial with coefficients in a field is an extension of that field over which the polynomial factors into linear factors. ...

References

• Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-36857-1.