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21. Referring to Example 50.9, show that G(Q(V2, iv3)/Q(iv3)) = < Z,, +>.

21. Referring to Example 50.9, show that G(Q(V2, iv3)/Q(iv3)) = < Z,, +>.

Algebra for College Students
Algebra for College Students
10th Edition
ISBN:9781285195780
Author:Jerome E. Kaufmann, Karen L. Schwitters
Publisher:Jerome E. Kaufmann, Karen L. Schwitters
Chapter10: Exponential And Logarithmic Functions
Section10.6: Exponential Equations, Logarithmic Equations, And Problem Solving
Problem 63PS
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Section 50, Number 21

21. Referring to Example 50.9, show that G(Q(V2, iV3)/Q(iV3)) = < Z,, +>.
Transcribed Image Text:21. Referring to Example 50.9, show that G(Q(V2, iV3)/Q(iV3)) = < Z,, +>.
50.9 Example Let 2 be the real cube root of 2, as usual. Now x - 2 does not split in Q(/2), for QV2) < R and only one zero of x³ – 2 is real. Thus x³ – 2 factors in (Q(/2))[x] into V2 and an irreducible quadratic factor. The splitting field E of x³ – 2 a linear factor x over Q is therefore of degree 2 over Q(V2). Then [E: Q] = [E : Q(2)][Q(/2) : Q] = (2)(3) = 6. We have shown that the splitting field over Q of x-2 is of degree 6 over Q. We can verify by cubing that and are the other zeros of x'-2 in C. Thus the splitting field E of x-2 over Q is Q/2, i/3). (This is not the same field as Q(2, i, V3), which is of degree 12 over Q.) Further study of this interesting example is left to the exercises (see Exercises 7, 8, 9, 16, 21, and 23). |
Transcribed Image Text:50.9 Example Let 2 be the real cube root of 2, as usual. Now x - 2 does not split in Q(/2), for QV2) < R and only one zero of x³ – 2 is real. Thus x³ – 2 factors in (Q(/2))[x] into V2 and an irreducible quadratic factor. The splitting field E of x³ – 2 a linear factor x over Q is therefore of degree 2 over Q(V2). Then [E: Q] = [E : Q(2)][Q(/2) : Q] = (2)(3) = 6. We have shown that the splitting field over Q of x-2 is of degree 6 over Q. We can verify by cubing that and are the other zeros of x'-2 in C. Thus the splitting field E of x-2 over Q is Q/2, i/3). (This is not the same field as Q(2, i, V3), which is of degree 12 over Q.) Further study of this interesting example is left to the exercises (see Exercises 7, 8, 9, 16, 21, and 23). |
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