Specific proof of irreducibility of $X^2-3$ in $\mathbb{Q}(\sqrt{2})[X]$

Question

I am working on the following exercise:

Prove that there exist no $a,b\in\mathbb{Q}$ such that $(a+b\sqrt{2})^2=3$. Conclude from this that $X^2-3$ is irreducible in $\mathbb{Q}(\sqrt{2})[X]$.

Proving the first part took me no effort. I am completely stuck on the second part, however. That is, how does the second part follow from the first part? Any help or hints would be appreciated!

Answer 1

Hint:

How do you characterize that a polynomial has root $\alpha$ in terms of divisibility?

Answer 2

Hint Assume by contradiction that $$X^2-3=(ax+b)(cx+d) $$ for some $a,b,c,d \in \mathbb Q(\sqrt{2})$.

Show that you have either $$a\sqrt{3}+b=0 \mbox{ or } \\ c\sqrt{3}+d=0 \\ $$

Deduce that $\sqrt{3}$ is a ratio of elements in $\mathbb Q(\sqrt{2})$ and hence an element in $\mathbb Q(\sqrt{2})$.