{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 320

# College Algebra Exam Review 320 - 330 7 FIELD EXTENSIONS...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 330 7. FIELD EXTENSIONS – FIRST LOOK The structure of the splitting ﬁeld can be better understood if we consider the possible intermediate ﬁelds between K and E , and introduce symmetry into the picture as well. Proposition 7.4.2. Let K be a ﬁeld, let f .x/ 2 KŒx be irreducible, and suppose ˛ and ˇ are two roots of f .x/ in some extension ﬁeld L. Then there is an isomorphism of ﬁelds W K.˛/ ! K.ˇ/ such that .k/ D k for all k 2 K and .˛/ D ˇ . Proof. According to Proposition 7.3.6, there is an isomorphism ˛ W KŒx=.f .x// ! K.˛/ that takes Œx to ˛ and ﬁxes each element of K . So the desired isomorphism K.˛/ Š K.ˇ/ is ˇ ı ˛ 1 . I Applying this result to the cubic equation, we obtain the following: For any two roots ˛i and ˛j of the irreducible cubic polynomial f .x/, there is an isomorphism K.˛i / Š K.˛j / that ﬁxes each element of K and takes ˛i to ˛j . Now, suppose that ı 2 K , so dimK .E/ D 3. Then E D K.˛i / for each i , so for any two roots ˛i and ˛j of f .x/ there is an automorphism of E (i.e., an isomorphism of E onto itself) that ﬁxes each element of K and takes ˛i to ˛j . Let us consider an automorphism of E that ﬁxes K pointwise and maps ˛1 to ˛2 . What is .˛2 /? The following general observation shows that .˛2 / is also a root of f .x/. Surely, .˛2 / ¤ ˛2 , so .˛2 / 2 f˛1 ; ˛3 g. We are going to show that necessarily .˛2 / D ˛3 and .˛3 / D ˛1 . Proposition 7.4.3. Suppose K Â L is any ﬁeld extension, f .x/ 2 KŒx, and ˇ is a root of f .x/ in L. If is an automorphism of L that leaves F ﬁxed pointwise, then .ˇ/ is also a root of f .x/. Proof. If f .x/ D 0: P fi x i , then P fi .ˇ/i D P . fi ˇ i / D .0/ D I The set of all automorphisms of a ﬁeld L, denoted Aut.L/, is a group. If F Â L is a subﬁeld, an automorphism of L that leaves F ﬁxed pointwise is called a F –automorphism of L. The set of F automorphisms of L, denoted AutF .L/, is a subgroup of Aut.L/ (Exercise 7.4.4). ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online