Unformatted text preview: 7.5. SPLITTING FIELDS OF POLYNOMIALS IN C Œx 339 Proof. E is also the splitting ﬁeld of f over M , so the previous result
applies with K replaced by M .
Now, we consider a converse:
Proposition 7.5.6. Suppose K Â E Â C are ﬁelds, dimK .E/ is ﬁnite,
and Fix.AutK .E// D K .
(a) For any ˇ 2 E , ˇ is algebraic over K , and the minimal polynomial for ˇ over K splits in EŒx.
(b) For ˇ 2 E , let ˇ D ˇ1 ; : : : ; ˇr be a list of the distinct elements
of f .ˇ/ W 2 AutK .E/g. Then .x ˇ1 /.: : : /.x ˇr / is the
minimal polynomial for ˇ over K .
(c) E is the splitting ﬁeld of a polynomial in KŒx. Proof. Since dimK .E/ is ﬁnite, E is algebraic over K .
Let ˇ 2 E , and let p.x/ denote the minimal polynomial of ˇ over K .
Let ˇ D ˇ1 ; : : : ; ˇr be the distinct elements of f .ˇ/ W 2 AutK .E/g.
Deﬁne g.x/ D .x ˇ1 /.: : : /.x ˇr / 2 EŒx. Every 2 AutK .E/ leaves
so the coefﬁcients of g.x/ lie in
Fix.AutK .E// D K . Since ˇ is a root of g.x/, it follows that p.x/ divides g.x/. On the other hand, every root of g.x/ is of the form .ˇ/
for 2 AutK .E/ and, therefore, is also a root of p.x/. Since the roots
of g.x/ Q simple (i.e., each root ˛ occurs only once in the factorization
g.x/ D .x ˛ /), it follows that g.x/ divides p.x/. Hence p.x/ D g.x/.
In particular, p.x/ splits into linear factors over E . This proves parts (a)
Since E is ﬁnite–dimensional over K , it is generated over K by ﬁnitely
many algebraic elements ˛1 ; : : : ; ˛s . It follows from part (a) that E is the
splitting ﬁeld of f D f1 f2 fs , where fi is the minimal polynomial of
˛i over K .
I Deﬁnition 7.5.7. A ﬁnite–dimensional ﬁeld extension K Â E Â C is said
to be Galois if Fix.AutK .E// D K .
With this terminology, the previous results say the following:
Theorem 7.5.8. For ﬁelds K Â E Â C , with dimK .E/ ﬁnite, the following are equivalent:
(a) The extension E is Galois over K . ...
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