Splitting field of a polynomial
From Encyclopedia of Mathematics
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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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=16268
Splitting field of a polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Splitting_field_of_a_polynomial&oldid=16268
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article