Prove sets have the same cardinality: Q(sqrt[5]) and Q

[Baron]

Prove \(\displaystyle |Q(\sqrt{5})|=|Q|\) where \(\displaystyle Q(\sqrt{5})=\{a+b\sqrt{5}:a,b\in Q\}\)

My work

Let \(\displaystyle A=Q(\sqrt{5})\) and \(\displaystyle B =Q\).

Then \(\displaystyle |B|\le|A|\) since \(\displaystyle B\subset A\)

So I only need to show there is an injection from \(\displaystyle |A|\) to \(\displaystyle |B|\) to conclude \(\displaystyle |A|\le|B|\) . I'm having difficulties showing there an injection exists.

 

[stapel]

Prove \(\displaystyle |Q(\sqrt{5})|=|Q|\) where \(\displaystyle Q(\sqrt{5})=\{a+b\sqrt{5}:a,b\in Q\}\)

My work

Let \(\displaystyle A=Q(\sqrt{5})\) and \(\displaystyle B =Q\).

Then \(\displaystyle |B|\le|A|\) since \(\displaystyle B\subset A\)

So I only need to show there is an injection from \(\displaystyle |A|\) to \(\displaystyle |B|\) to conclude \(\displaystyle |A|\le|B|\) . I'm having difficulties showing there an injection exists.

Maybe something like they show on page 2 of this PDF document?