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21, Referring to Example 50.9, show that G(Q&2, i/3)/Q(i 3) ~ (Za, +).

21, Referring to Example 50.9, show that G(Q&2, i/3)/Q(i 3) ~ (Za, +).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 58E
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Section 50 number 21
50.9 Example Let 2 be the real cube root of 2, as usual. Now x'-2 does not split in Q(/2), for <R and only one zero of x3 - 2 is real. Thus x³ - 2 factors in (Q(/2))[x]into. a linear factor x – 2 and an irreducible quadratic factor. The splitting field E of x³ – 2 over Q is therefore of degree 2 over Q2). 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, wi16, 21, and 23). bionduenoino ind
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 <R and only one zero of x3 - 2 is real. Thus x³ - 2 factors in (Q(/2))[x]into. a linear factor x – 2 and an irreducible quadratic factor. The splitting field E of x³ – 2 over Q is therefore of degree 2 over Q2). 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, wi16, 21, and 23). bionduenoino ind
20. Show that Q02) has enly the identity automorphism> 21, Referring to Example 50.9, show that G(Q(/2, iv3)/Q(i V3) ~ (Z3, +).
Transcribed Image Text:20. Show that Q02) has enly the identity automorphism> 21, Referring to Example 50.9, show that G(Q(/2, iv3)/Q(i V3) ~ (Z3, +).
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