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3. Given a prime number k, we define Q(√k) = {a+b√k : a, b ≤ Q} ≤ R. This set becomes a field when equipped with the usual addition and multiplication operations inherited from R. (a) For each non-zero x € Q(√2) of the form x = a +b√√2, prove that x-¹ b = a²-26² a²-26² √2. (b) Show that √√2 Q(√3). You can use, without proof, the fact that √2, √3, are all irrational numbers. (c) Show that there cannot be a function : Q(√2)→ Q(√3) so that 6: (Q(√2) - {0}, ×) → (Q(√3) − {0}, ×) and 6: (Q(√2), +) → (Q(√3), +) are both group isomorphisms. Hint: What can you say about $(√2 × √√2)?

3. Given a prime number k, we define Q(√k) = {a+b√k : a, b ≤ Q} ≤ R. This set becomes a field when equipped with the usual addition and multiplication operations inherited from R. (a) For each non-zero x € Q(√2) of the form x = a +b√√2, prove that x-¹ b = a²-26² a²-26² √2. (b) Show that √√2 Q(√3). You can use, without proof, the fact that √2, √3, are all irrational numbers. (c) Show that there cannot be a function : Q(√2)→ Q(√3) so that 6: (Q(√2) - {0}, ×) → (Q(√3) − {0}, ×) and 6: (Q(√2), +) → (Q(√3), +) are both group isomorphisms. Hint: What can you say about $(√2 × √√2)?

Advanced Engineering Mathematics
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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3. Given a prime number k, we define Q(√k) = {a+b√k : a,b ≤ Q} ≤ R. This set becomes a field when equipped with the usual addition and multiplication operations inherited from R. (a) For each non-zero x = Q(√2) of the form x = a +b√√/2, prove that x-¹ a = a²26² b a²-26² √2. (b) Show that √2 Q(√3). You can use, without proof, the fact that √2,√√3, are all irrational numbers. (c) Show that there cannot be a function : Q(√2)→ Q(√3) so that 6: (Q(√2) - {0}, ×) → (Q(√3) − {0}, ×) and 6: (Q(√2), +) → (Q(√3), +) are both group isomorphisms. Hint: What can you say about (√2 × √√2)?
Transcribed Image Text:3. Given a prime number k, we define Q(√k) = {a+b√k : a,b ≤ Q} ≤ R. This set becomes a field when equipped with the usual addition and multiplication operations inherited from R. (a) For each non-zero x = Q(√2) of the form x = a +b√√/2, prove that x-¹ a = a²26² b a²-26² √2. (b) Show that √2 Q(√3). You can use, without proof, the fact that √2,√√3, are all irrational numbers. (c) Show that there cannot be a function : Q(√2)→ Q(√3) so that 6: (Q(√2) - {0}, ×) → (Q(√3) − {0}, ×) and 6: (Q(√2), +) → (Q(√3), +) are both group isomorphisms. Hint: What can you say about (√2 × √√2)?
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