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**Definition 6.5.1.**
Let K be a field and let
f(x) = a_{0} + a_{1} x
+ **· · ·** +
a_{n}x^{n}
be a polynomial in K[x] of degree n>0.
An extension field F of K is called a
**splitting field for f(x) over K**
if there exist elements
r_{1},
r_{2},
. . . ,
r_{n} in F such that

**(i)**f(x) = a_{n}(x-r_{1}) (x-r_{2})**· · ·**(x-r_{n}), and**(ii)**F = K(r_{1},r_{2},...,r_{n}).

In the above situation we usually say that f(x)
**splits**
over the field F. The elements
r_{1},
r_{2},
. . . ,
r_{n}
are roots of f(x), and so F is obtained
by adjoining to K a complete set of roots of f(x).

**Theorem 6.4.2.**
Let f(x) be a polynomial in K[x] of degree n>0.
Then there exists a splitting field F for f(x) over K, with
[F:K] n!.

**Lemma 6.4.3.**
Let
: K -> L
be an isomorphism of fields.
Let F be an extension field of K such
that F = K(u) for an algebraic element u in F.
Let p(x) be the minimal polynomial of u over K.
If v is any root of the image q(x) of p(x) under
,
and E=L(v), then there is a unique way to extend
to an isomorphism
: F -> E such that
(u) = v and
(a) =
(a)
for all a in K.

**Theorem 6.4.5.**
Let f(x) be a polynomial over the field K.
The splitting field of f(x) over K is unique up to isomorphism.

**1.**
Find the splitting field over **Q** for the polynomial
x^{4} + 4.
*Solution*

**2.**
Find the degree of the splitting field over **Z**_{2}
for the polynomial (x^{3} + x + 1)(x^{2} + x + 1).
*Solution*

**3.**
Find the degree [F:**Q**],
where F is the splitting field of the polynomial
x^{3} - 11
over the field **Q** of rational numbers.
*Solution*

**4.**
Determine the splitting field over **Q** for x^{4} + 2.
*Solution*

**5.**
Determine the splitting field over **Q** for
x^{4} + x^{2} + 1.
*Solution*

**6.**
Factor x^{6} - 1 over **Z**_{7};
factor x^{5} - 1 over **Z**_{11}.
*Solution*

Solutions to the problems | Forward to §6.5 | Back to §6.3 | Up | Table of Contents