Quadratic field extension of the rational number

• March 22nd 2011, 09:08 AM
nyammo
Quadratic field extension of the rational number
about the set of number W={a+(2^(1/2))b: a,b∈Q}
Q is set of rational number.

a)how to show x=a+(2^(1/2))b∈Q if and only if b=0?
b)how to show W is countable?
c)how to show that between any two distinct real number is an element of W?
d)how to show W is an ordered field?
• March 22nd 2011, 10:30 AM
FernandoRevilla
Two little hints:

a) Prove that if $a+b\sqrt{2}\in \mathbb{Q}$ and $b\neq 0$ then, $\sqrt{2}$ would be rational (contradiction).

b) $\mathbb{Q}$ is countable and $\#(W)=\#(\mathbb{Q}\times \mathbb{Q})$ .

Try the rest.