Refer to Example 50.9 for Exercises 7 through 9. 7. What is the order of G(Q(2)/Q)? 8. What is the order of G(Q(/2, i/3)/Q)?
Refer to Example 50.9 for Exercises 7 through 9. 7. What is the order of G(Q(2)/Q)? 8. What is the order of G(Q(/2, i/3)/Q)?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Section 50 number 8 and 7
![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).
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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
Transcribed Image Text:Refer to Example 50.9 for Exercises 7 through 9.
7. What is the order of G(Q(2)/Q)?
8. What is the order of G(Q(/2, i/3)/Q)?
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