3. (16 marks) 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 -
a
a+b√2, prove that x-1
b
=
²-26²
a²26² √2.
-262
(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}, ×) → (Q(√3) − {0}, ×)
and
6 : (Q(√2), +) → (Q(√3), +)
are both group isomorphisms. Hint: What can you say about (√2 × √2)?
