Splitting Field
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K over which P factorizes ("splits", hence the name of a splitting field) into linear factors X ai, and such that the ai 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 t...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K over which P factorizes ("splits", hence the name of a splitting field) into linear factors X ai, and such that the ai 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). For an example if K is the rational number field Q and P(X) = X3 2, then a splitting field L will contain a primitive cube root of unity, as well as a cube root of 2.
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