Unformatted text preview: 337 7.5. SPLITTING FIELDS OF POLYNOMIALS IN C Œx E 2 2 K.˛1 / r
d K.˛2 / K.˛3 / d d 3 d
d Q ¨ 3 ¨¨ 2 ¨¨ ¨
¨¨ K.ı/ ¨¨ Figure 7.4.2. Intermediate ﬁelds for a splitting ﬁeld with Galois group S3 . 7.5. Splitting Fields of Polynomials in C Œx
Our goal in this section will be to state a generalization of Theorem 7.4.9
for arbitrary polynomials in KŒx for K a subﬁeld of C and to sketch some
ideas involved in the proof. This material will be treated systematically and
in a more general context in the next chapter.
Let K be any subﬁeld of C , and let f .x/ 2 KŒx. According to
Gauss’s fundamental theorem of algebra, f .x/ factors into linear factors
in C Œx. The smallest subﬁeld of C that contains all the roots of f .x/
in C is called the splitting ﬁeld of f .x/. As for the cubic polynomial,
in order to understand the structure of the splitting ﬁeld, it is useful to
introduce its symmetries over K . An automorphism of E is said to be a
K -automorphism if it ﬁxes every point of K . The set of K –automorphisms
forms a group (Exercise 7.4.4).
Here is the statement of the main theorem concerning subﬁelds of a
splitting ﬁeld and the symmetries of a splitting ﬁeld:
Theorem 7.5.1. Suppose K is a subﬁeld of C , f .x/ 2 KŒx, and E is the
splitting ﬁeld of f .x/ in C .
(a) dimK .E/ D jAutK .E/j.
(b) The map M 7! AutM .E/ is a bijection between the set of intermediate ﬁelds K Â M Â E and the set of subgroups of
AutK .E/. The inverse map is H 7! Fix.H /. In particular, there
are only ﬁnitely many intermediate ﬁelds between K and E . ...
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