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:
“Theories that go counter to the facts of human nature are foredoomed.”
—Edith Hamilton (1867–1963)
“A judge is not supposed to know anything about the facts of life until they have been presented in evidence and explained to him at least three times.”
—Parker, Lord Chief Justice (1900–1972)
“Obviously the facts are never just coming at you but are incorporated by an imagination that is formed by your previous experience. Memories of the past are not memories of facts but memories of your imaginings of the facts.”
—Philip Roth (b. 1933)