I am trying to understand quadratic integer rings. The following is assuming $d$ is square-free. 

From my understanding so far:

$\mathbb Z\big[\sqrt d\big]$ is an integral domain.

$\mathbb Q\big[\sqrt d\big]$ is a field.

$\mathbb Q\big[\sqrt d\big]$ is the field of fractions of $\mathbb Z\big[\sqrt d\big]$.

$\mathbb Z\big[\sqrt d\big]$ is the ring of quadratic integers when $d \equiv 2,3 \pmod 4$.

$\mathbb Z \left[\frac{1+\sqrt d}{2} \right]$ is the ring of quadratic integers when $d \equiv 1 \pmod 4$.

Why isn't $\mathbb Z\big[\sqrt d\big]$ a quadratic integer ring when $d \equiv 1 \pmod 4$?