**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:

“Let us not underrate the value of a fact; it will one day flower in a truth. It is astonishing how few *facts* of importance are added in a century to the natural history of any animal. The natural history of man himself is still being gradually written.”

—Henry David Thoreau (1817–1862)

“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)

“I went to the woods because I wished to live deliberately, to front only the essential *facts* of life, and see if I could not learn what it had to teach, and not, when I came to die, discover that I had not lived.... I wanted to live deep and suck out all the marrow of life, to live so sturdily and Spartan-like as to put to rout all that was not life, to cut a broad swath and shave close, to drive life into a corner, and reduce it to its lowest terms.”

—Henry David Thoreau (1817–1862)