Facts
An extension L which is a splitting field for multiple polynomials p(X) over K is called a normal extension.
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 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 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′.
Read more about this topic: Splitting Field
Famous quotes containing the word facts:
“One would appear ridiculous who would say, that it is only probable the sun will rise to-morrow, or that all men must die; though it is plain we have no further assurance of these facts than what experience affords us.”
—David Hume (1711–1776)
“There are in me, in literary terms, two distinct characters: one who is taken with roaring, with lyricism, with soaring aloft, with all the sonorities of phrase and summits of thought; and the other who digs and scratches for truth all he can, who is as interested in the little facts as the big ones, who would like to make you feel materially the things he reproduces.”
—Gustave Flaubert (1821–1880)
“To-day ... when material prosperity and well earned ease and luxury are assured facts from a national standpoint, woman’s work and woman’s influence are needed as never before; needed to bring a heart power into this money getting, dollar-worshipping civilization; needed to bring a moral force into the utilitarian motives and interests of the time; needed to stand for God and Home and Native Land versus gain and greed and grasping selfishness.”
—Anna Julia Cooper (1859–1964)