College Algebra Exam Review 320

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

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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  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  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). ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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