A splitting field is an extension field 
of a field 
for a polynomial ![f(x) in F[x]](/images/equations/SplittingField/Inline3.svg)
such that 
factors completely into linear factors in ![K[x]](/images/equations/SplittingField/Inline5.svg)
and 
does not factor completely into linear factors over any
proper subfield of 
containing 
(Dummit and Foote 2004, p. 536).
For example, the extension field 
is the splitting field for 
since it is the smallest field containing its roots, 
and 
. Note that it is also the splitting field for 
.
