Questions by yweimann - Page 58
Given a prime number k, we define Q(k) = {a+bk: a, b e Q) CR. This set becomes a field when equipped with the usual addition and multiplication operations inherited from IR. (a) For each non-zero a Q(2) of the form a = a +b2, prove that a -26-262 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 :(Q(2)(0), x) (Q(3) - {0}, ) and 6: (Q(2), +) (Q(3), +) are both group isomorphisms. Hint: What can you say about o(2 2)?