Let Q be the field of rational numbers and let F Q(√2, √7). You may assume that √2, √7, √14 are irrational. (a) Show that √7 Q(√2). Hence obtain that [F : Q] = 4.
Let Q be the field of rational numbers and let F Q(√2, √7). You may assume that √2, √7, √14 are irrational. (a) Show that √7 Q(√2). Hence obtain that [F : Q] = 4.
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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I don't understand 2 points in the solution for this problem. Can you please explain why? Thanks.
1. Why : {1, √2} is a basis for Q(√2) over Q then we conclude 2rs = 0 and r2 + 2s2 = 0 ???
2. Why: [F : Q(√2) ] = 2
![Let be the field of rational numbers and let F = Q(√2, √7). You may assume that
√2, √7, √14 are irrational.
(a) Show that √7 ¢ Q(√2). Hence obtain that [F : Q] = 4.
Answer
=
(a) Assume that √7 € Q(√2). Then there are r, s € Q such that √7 = r + s√√2. Thus
7 = r² +2s² +2rs√2. Since {1, √2} is a basis for Q(√2) over Q we obtain that 2rs = 0
and r² + 2s² 7. Therefore r = 0, 2s²
7, or s = 0, p²
7. The former implies
(2s)² = 14. In either case we have a contradiction because √14 and √7 are irrational.
By the degree formula [F : Q] = [F : Q(√2)] · [Q(√2) : Q] = 2 ⋅ 2 = 4.
=
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09c2f836-a777-444b-9271-94b360db7692%2F15dd1d86-c038-4183-9490-65995f0564f6%2Fsb9ff779_processed.png&w=3840&q=75)
Transcribed Image Text:Let be the field of rational numbers and let F = Q(√2, √7). You may assume that
√2, √7, √14 are irrational.
(a) Show that √7 ¢ Q(√2). Hence obtain that [F : Q] = 4.
Answer
=
(a) Assume that √7 € Q(√2). Then there are r, s € Q such that √7 = r + s√√2. Thus
7 = r² +2s² +2rs√2. Since {1, √2} is a basis for Q(√2) over Q we obtain that 2rs = 0
and r² + 2s² 7. Therefore r = 0, 2s²
7, or s = 0, p²
7. The former implies
(2s)² = 14. In either case we have a contradiction because √14 and √7 are irrational.
By the degree formula [F : Q] = [F : Q(√2)] · [Q(√2) : Q] = 2 ⋅ 2 = 4.
=
=
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