Splitting field

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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 into linear factors

Xai,

and such that the ai generate L over K. 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, anyway).

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.

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.

Therefore, for example, for K given as 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 a vicious circle.

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′.

Examples

  • The splitting field of x2 + 1 over R, the real numbers, is C, the complex numbers.
  • The splitting field of x2 + 1 over GF7 is GF72.
  • The splitting field of x2 − 1 over GF7 is GF7 since x2 − 1 = (x + 1)(x − 1) already factors into linear factors.

See also

References

  • Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-36857-1.