3.Given a prime number k, we define Q(√k) = {a+b√k : a,b ≤ Q} ≤ R. This setbecomes a field when equipped with the usual addition and multiplication operations inheritedfrom R.(a) For each non-zero x = Q(√2) of the form x = a +b√√/2, prove that x-¹a=a²26²ba²-26²√2.(b) Show that √2 Q(√3). You can use, without proof, the fact that √2,√√3,are allirrational numbers.(c) Show that there cannot be a function : Q(√2)→ Q(√3) so that6: (Q(√2) - {0}, ×) → (Q(√3) − {0}, ×)and6: (Q(√2), +) → (Q(√3), +)are both group isomorphisms. Hint: What can you say about (√2 × √√2)?

Answered: Image /qna-images/answer/87682e9f-aec2-424c-afdb-c1fedbefb9e8.jpg

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)?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.6: Congruence Classes

Problem 9E: Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[...

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