I am solving an exercise with the field extension $\mathbb{Q}(\sqrt5)/\mathbb{Q}$
I am suck trying to prove that:$$B = \{a+b(\dfrac{1+\sqrt5}{2}) |a,b\in \mathbb{Z} \}$$ Where B is the integer ring of $\mathbb{Q}(\sqrt5)$.
The deffinition I have of integer ring is: the subset of elements of $\mathbb{Q}(\sqrt5)$ whose irreducible polynomial in $\mathbb{Q}$ has integer coefficients.
I have showed that $B \supset \{a+b(\dfrac{1+\sqrt5}{2}) |a,b\in \mathbb{Z} \}$, but I cannot fully prove the equality
I have read that $\{ 1, \dfrac{1+\sqrt5}{2}\}$ is a basis of B, but I haven't found a proof.
If someone could porvide a proof, give me a hint or point me to some source material I would be very thankful.