The formula for the Minkowski bound is $$M=(\frac{4}{\pi})^{s}\frac{n!}{n^n}\left| \Delta \right|^{\frac{1}{2}}$$ where $s$ is the number of pairs of complex embeddings. In our case, $s=0$, $n=2$, $\Delta=32$,
so $M$ shouldn't be $\frac{2}{4}\sqrt{32}=\frac{2}{4}4\sqrt{2}=2\sqrt{2}$?
But sagemath said $M$ should be $\sqrt{2}$ using the code:
x = polygen(ZZ)
f=x^2 - 8
K.< c > =NumberField(f)
M=K.minkowski_bound()
print(M)
Where did I do wrong?
No, $\Delta$ is not $32$. You need to verify that the radicand is square-free and, if not, divide by all the squared factors until you find something that is. This is the reduced radicand from which you get the proper value for the discriminant.
To get this reduced radicand you can render the product of all distinct prime factors with an odd exponent and take each such factor only once. Here, $8=2^3$ where there is only the prime factor $2$ and it has an odd exponent, so the radicand reduces to just $2$ (and thus $\mathbb Q[\sqrt8]=\mathbb Q[\sqrt2]$). Now apply the formula for the discriminant, which gives $8$, and plug into the Minkowski bound.