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