3.2. SPLITTING FIELDS 5
Problems For Section 3.1
1. Let E be an extension of F , and let S be a subset of E.IfF(S) is the subfield of E
generated by S over F , in other words, the smallest subfield of E containing F and S,
describe F (S) explicitly, and justify your characterization.
2. If for each i ∈ I, K
i
is a subfield of the field E, the composite of the K
i
(notation
i
K
i
) is the smallest subfield of E containing every K
i
. As in Problem 1, describe
the composite explicitly.
3. Assume that α is algebraic over F , with [F [α]:F ]=n.Ifβ ∈ F [α], show that
[F [β]:F ] ≤ n, in fact [F [β]:F ] divides n.
4. The minimal polynomial of
√
2 over the rationals Q is X
2
−2, by (3) of (3.1.6). Thus
Q[
√
2] consists of all numbers of the form a
0
+ a
1
√
2, where a
0
and a
1
are rational.
By Problem 3, we know that −1+
√
2 has a minimal polynomial over Q of degree at
most 2. Find this minimal polynomial.
5. If α is algebraic over F and β belongs to F [α], describe a systematic procedure for
finding the minimal polynomial of β over F .
6. If E/F and the element α ∈ E is transcendental over F , show that F (α) is isomorphic
to F (X), the field of rational functions with coefficients in F .
7. Theorem 3.1.3 gives one method of adjoining a root of a polynomial, and in fact there
is essentially only one way to do this. If E is an extension of F and α ∈ E is algebraic
over F with minimal polynomial m(X), let I be the ideal m(X)⊆F [X]. Show that
F (α) is isomorphic to F [X]/I. [Define ϕ : F [X] → E by ϕ(f(X)) = f(α), and use
the first isomorphism theorem for rings.]
8. In the proof of (3.1.3), we showed that if f is irreducible in F [X], then I = f is a
maximal ideal. Show that conversely, if I is a maximal ideal, then f is irreducible.
9. Suppose that F ≤ E ≤ L, with α ∈ L. What is the relation between the minimal
polynomial of α over F and the minimal polynomial of α over E?
10. If α
1
,...,α
n
are algebraic over F , we can successively adjoin the α
i
to F to obtain
the field F [α
1
,...,α
n
] consisting of all polynomials over F in the α
i
. Show that
[F [α
1
,...,α
n
]:F ] ≤
n
i=1
[F (α
i
):F ] < ∞
3.2 Splitting Fields
If f is a polynomial over the field F , then by (3.1.3) we can find an extension E
1
of F
containing a root α
1
of f . If not all roots of f lie in E
1
, we can find an extension E
2
of E
1
containing another root α
2
of f. If we continue the process, eventually we reach a
complete factorization of f. In this section we examine this idea in detail.
If E is an extension of F and α
1
,...,α
k
∈ E, we will use the notation F(α
1
,...,α
k
)
for the subfield of E generated by F and the α
i
.ThusF (α
1
,...,α
k
) is the smallest
subfield of E containing all elements of F along with the α
i
. ( “Smallest” means that
F (α
1
,... ,α
k
) is the intersection of all such subfields.) Explicitly, F (α
1
,...,α
k
)isthe
collection of all rational functions in the α
i
with nonzero denominators.
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