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

and such that the coefficients *a*_{i} 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 to be the Galois group of *p* (if we assume it is separable).

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### Famous quotes containing the words splitting and/or field:

“I had an old axe which nobody claimed, with which by spells in winter days, on the sunny side of the house, I played about the stumps which I had got out of my bean-field. As my driver prophesied when I was plowing, they warmed me twice,—once while I was *splitting* them, and again when they were on the fire, so that no fuel could give out more heat. As for the axe,... if it was dull, it was at least hung true.”

—Henry David Thoreau (1817–1862)

“My business is stanching blood and feeding fainting men; my post the open *field* between the bullet and the hospital. I sometimes discuss the application of a compress or a wisp of hay under a broken limb, but not the bearing and merits of a political movement. I make gruel—not speeches; I write letters home for wounded soldiers, not political addresses.”

—Clara Barton (1821–1912)