Prove that $\{1, \alpha, \alpha^{2}\}$ is an integral basis of $\mathcal O_{K}$ Let $K = \mathbb{Q}(\alpha)$ where $\alpha^3 - 50\alpha- 10= 0$.
Prove that $\{1, \alpha, \alpha^{2}\}$ is an integral basis of $\mathcal O_{K}$.
I know that the minimal polynomial is

$$m_\alpha(x)=x^3 - 50x -10$$

but I'm not sure where to even begin. I have looked at plenty of resources but none of them seem to have concrete examples of how to solve a problem like this. I've tried to understand general examples but I'm not sure how to solve a specific problem like this one. Thanks in advance.
 A: Eisenstein easily verifies the polynomial is irreducible. Next, consider that the discriminant of the polynomial $x^3+ax+b$ is $-4a^3-27b^2=497300$ which is $100$ times a prime. So if we can show that $2$ and $5$ are ramified, we will see that this is an integral basis as otherwise the true discriminant would differ from ours by dividing by $4, 25,\,$ or $100$ using Dedekind's theorem on ramification.
But then this is simple as reducing $x^3-50x-10$ mod $2$ and $5$ both give $x^3$ (and both $2$ and $5$ only divide the constant term once each see theorem 3.1 for example) so we see that $(2)$ and $5$ are totally ramified as $(2,\alpha)^3, (5,\alpha)^3$ respectively, so $10|\Delta_K$, which is what we set out to show.
As KCd notes in the comments, you an also end this a bit earlier by appealing to theorem 2.3 in the linked notes from above. I usually prefer the full ramification information myself, because that's the form I usually use things in, but in your case if all you care about is showing it's a power basis you can save a little time by going straight there.
A: First, compute the discriminant of $1,\alpha,\alpha^2$  (I assume you know how to do that?).  That's $2^2 5^2  4973$. Next, the discriminant of ${\cal O}_K$ must divide that, and the quotient must be a square. However, the polynomial is 2-Eisenstein as well as 5-Eisenstein, which implies that $2$ and $5$ ramify, which implies that they divide the discriminant of ${\cal O}_K$. That leaves only 1 possibility for the discriminant of ${\cal O}_K$, namely, it has to be $2^2 5^2  4973$ as well. Hence $1,\alpha,\alpha^2$ is a $\mathbb{Z}$-basis of ${\cal O}_K$.
For more complicated examples you should simply use a computer algebra system (Sage, Magma, Maple, etc.) to compute the integral basis. Trying to do this by hand makes little sense. If you want to know how those programs work, there are various methods, some of which are not too difficult to understand (e.g. there is a book by Cohen on computational number theory, and there is also an online book by William Stein).
