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Self taught here so please bear with me. How does one define the ring of integers of the field $\mathbb{Q}(r)$, where $r$ is a root of the cubic $$x^3+px+q$$ as well as determining the fundamental units of this field, alongside determining which primes are irreducible and reducible (assuming it's a principal ideal domain).

If it helps, assume I know how to calculate the coefficients of the minimal polynomial to an element of the field with respect to whatever the basis is.

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    $\begingroup$ The definition of the ring of integers in a number field $K$ is always the same, independently of what the number field is: it is the subring of $K$ of those elements which are roots of a monic polynomial with integer coefficients. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 5, 2016 at 5:46
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    $\begingroup$ It may be the case that what you are really after is not how one defines the ring of integers but how one computes it (in that case, you should edit the question so that it asks what you really want). If you search this site, you will find several examples of that computation being done. The general case, though, is somewhat complicated. It was worked out by A. A. Albert in 1930 in this paper, to which you may or may not have access. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 5, 2016 at 5:52
  • $\begingroup$ For fun, @MarianoSuárez-Álvarez, when I arrived at Brown, Albert was still alive, an emeritus member of our department, and I actually met him. $\endgroup$
    – Lubin
    Commented Sep 5, 2016 at 23:57
  • $\begingroup$ @Lubin, he is one of my heroes. I've spent many hours reading in wild amazement his collected papers :-) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Sep 6, 2016 at 1:30
  • $\begingroup$ @MarianoSuárez-Álvarez, I must retract everything. I looked up Albert’s obituary, and he was never at Brown. So I must have met some other (much more) elderly mathematician and confused the two. $\endgroup$
    – Lubin
    Commented Sep 6, 2016 at 3:00

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In complete agreement with Mariano Suárez-Álvarez’s comment, I would suggest that if you have a specific case of a cubic, and $\lambda$ is a root, look at the minimal polynomial of $A+B\lambda+C\lambda^2$, and see whether, by guess and by golly, you can figure out the complete criteria for this general element of $\Bbb Q(\lambda)$ to be an algebraic integer. In my experience, this can be the very most difficult part of the task.

Then, believe it or not, finding the class group is comparatively easy. But even when you know (as is the case when the field has one real archimedean absolute value and one (pair of) complex, so that there is only one primitive unit), finding a basis for the units can be extremely difficult with only elementary methods.

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