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:
“Modern morality and manners suppress all natural instincts, keep people ignorant of the facts of nature and make them fighting drunk on bogey tales.”
—Aleister Crowley (1875–1947)
“Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!”
—Charles Dickens (1812–1870)
“Neither moral relations nor the moral law can swing in vacuo. Their only habitat can be a mind which feels them; and no world composed of merely physical facts can possibly be a world to which ethical propositions apply.”
—William James (1842–1910)