In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors.
A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors
where and for each we have with ai not necessarily distinct and such that the roots ai generate L over K. The extension L is then an extension of minimal degree over K in which p splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of p (if we assume it is separable).
An extension L which is a splitting field for a set of polynomials p(X) over K is called a normal extension of K.
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. If K is a subfield of the complex numbers, the existence is immediate. On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid circular reasoning.
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′.
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as x2 + 1 over R, the real numbers, have no roots. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
Let F be a field and p(X) be a polynomial in the polynomial ring F[X] of degree n. The general process for constructing K, the splitting field of p(X) over F, is to construct a chain of fields such that Ki is an extension of Ki −1 containing a new root of p(X).
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In field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients.
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