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Splitting field of a polynomial

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The smallest field containing all roots of that polynomial. More exactly, an extension of a field is called the splitting field of a polynomial over the field if decomposes over into linear factors:

and if (see Extension of a field). A splitting field exists for any polynomial , and it is defined uniquely up to an isomorphism that is the identity on . It follows from the definition that a splitting field is a finite algebraic extension of .

Examples. The field of complex numbers serves as the splitting field of the polynomial over the field of real numbers. Any finite field , where , is the splitting field of the polynomial over the prime subfield .


Comments

See also Galois theory; Irreducible polynomial.

References

[a1] I. Stewart, "Galois theory" , Chapman & Hall (1979)
How to Cite This Entry:
Splitting field of a polynomial. O.A. Ivanova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=16268
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098