Splitting Field

The extension field K of a field F is called a splitting field for the polynomial f(x) in F[x] if f(x) factors completely into linear factors in K[x] and f(x) does not factor completely into linear factors over any proper subfield of K containing F (Dummit and Foote 1998, p. 448).

For example, the extension field Q(sqrt(3)i) is the splitting field for x^2+3 since it is the smallest field containing its roots, sqrt(3)i and -sqrt(3)i. Note that it is also the splitting field for x^3+1.

Portions of this entry contributed by Todd Rowland

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Dummit, D. S. and Foote, R. M. Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1998.

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Splitting Field

Cite this as:

Rowland, Todd and Weisstein, Eric W. "Splitting Field." From MathWorld--A Wolfram Web Resource.

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