Question

10. Use 9 above to prove that the equation x^2 − 2y^2 = 1 has infinitely many solutions over Q. What can

you conclude about the number of solutions over Z?

(question9:  For F as in 8, define N : F → Q by N(a + b√2) = a^2 − 2b^2.

(i) Prove that N(αβ) = N(α)N(β), for all α,β ∈ F.
(ii) Find an element u ∈ F such that N(u) = 1 and such that all of the powers u^n are distinct elements

of F.)

(question 8

Let F denote the set of real numbers of the form a+b√2, with a, b∈Q.

(i) Show that F is a subfield of R.

(ii) Prove that a+b√2=c+d√2 if and only if a=c and b=d.)

Answer 1

O. Given-that use q above to poe ttat, the eo X.25% has fnfinitely mony Soludtions ovea F is Subfield of IR Define N: -

2 have to find am element ue F Such that i)