Question: 3. (16 marks) Given a prime number k, we dene (Xx/E) = {a + bx/E : a, b E Q} g IR. This set becomes

3. (16 marks) Given a prime number k, we dene (Xx/E) = {a + bx/E : a, b E Q} g IR. This set becomes a eld when equipped with the usual addition and multiplication operations inherited from IR. (a) For each nonzero as E (Xx/5) of the form as = a + bx/, prove that 531 = . (b) Show that x/i Q(\\/)- You can use, without proof, the fact that Mix/3, % are all irrational numbers. (0) Show that there cannot be a function : Q(\\/) > (Db/3) so that = (Q(\\/) _ {0}7 X) _) (Q('/) _ {0}7 X) and a3 = (@(m a (@(m are both group isomorphisms. Hint: What can you say about (\\/ x W)

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