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Splitting Field

A splitting field is an extension field K of a field F for a nonzero polynomial f(x) in F[x] such that 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 2004, p. 536).

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.

For an irreducible polynomial f in Q[x] of polynomial degree 3, let alpha be any root and let Delta be its polynomial discriminant. The splitting field is

K=Q(alpha,sqrt(Delta)).

Its extension field degree over Q is 3 if Delta is the square of a rational number and 6 otherwise (Milne 2022, p. 49). Thus adjoining one root need not give the splitting field.

See also

Extension Field, Galois Group, Primitive Element, Separable Polynomial

Portions of this entry contributed by Todd Rowland

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References

Dummit, D. S. and Foote, R. M. "Splitting Fields and Algebraic Closures." §13.4 in Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley, pp. 536-544, 2004.Milne, J. S. Fields and Galois Theory, Version 5.10. Sep. 2022. https://www.jmilne.org/math/CourseNotes/FT.pdf.Trott, M. "Solution of Equations." §1.5 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5. https://www.mathematicaguidebooks.org/.

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Splitting Field

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Splitting Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SplittingField.html

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