number fields with discriminant less than 8

Question

Given the discriminant $|d_k| \le 8$. Determine all possible number fields of degree $n$.

Using Minkowski I found the following bound on $n$ :

$$\sqrt{|d_k|} \ge (\frac{n^n}{n!})(\frac{\pi}{4})^{n/2}$$

which would give me that the only possible values for $n$ are $1$ or $2$.

How can I now determine all possible number-fields of degree $2$ and $d_k \le 8$ ?

Would appreciate any tips.

Answer 1

You know that your number field is of the form $\Bbb Q(\sqrt{m})$ for some square-free integer $m$.

It is not hard to check that $d_m = m$ if $m \equiv 1 \mod 4$ and $d_m = 4m$ otherwise.