I am working on the text book "algebraic number theory" by Jurgen Neukirch(P15, exercise 6). To prove the integer basis is$ \{1, \theta, \frac{\theta^2+\theta}{2}\}$. After a long and tedious calculation, I still get nothing.(Following the method which can be used in exercise 5: the $\mathbb{Q}(\sqrt[3]{2})$'s case, which is typical and you can find in stackexchange!).
Any help is going to be appreciated.
First of all, you have to show that $\frac{\theta^2+\theta}{2}$ is integral over $\mathbb{Z}$. Then, show that $1,\theta,\frac{\theta^2+\theta}{2}$ are linearly independent over $\mathbb{Z}$. Compute the discriminant $d(1,\theta,\frac{\theta^2+\theta}{2})$. The result will be -107 (at least according to what I have found out so far). Since 107 is a prime number and thus square-free, you can use Theorem 2.12, ch.1 of Neukirch's book to conclude that $1,\theta,\frac{\theta^2+\theta}{2}$ is a basis of the ring of integers of $\mathbb{Q}(\theta)$.