Timeline for Trace-free basis for $\mathcal{O}_K$, $K$ a cubic field

5 events

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Jul 10, 2022 at 18:46	comment	added	KConrad		To add some detail to Gerry's comment, if such $\alpha$ and $\beta$ exist then ${\rm Tr}_{K/\mathbf Q}(a + b\alpha + c\beta) = 3a$ for all integers $a$, $b$, and $c$, so ${\rm Tr}_{K./\mathbf Q}(\mathcal O_K) = 3\mathbf Z$ and thus we just need a cubic field containing an algebraic integer whose trace is not divisible by $3$.
Jul 7, 2022 at 3:14	comment	added	Gerry Myerson		This is very nice, as it gives a simple necessary condition for the existence of $\alpha$ and $\beta$. As an answer to the question of whether there always exist such $\alpha$ and $\beta$, bringing in the discriminant and different and inverse different is a bit of overkill, isn't it? All one needs is a monic, irreducible cubic with quadratic term not a multiple of three, such as $x^3-x^2-1$, isn't that right? Or, $x^3+x^2-2x-1$, if you want a cyclic example.
Jul 6, 2022 at 17:30	vote	accept	Stanley Yao Xiao
Jul 6, 2022 at 14:51	history	edited	Jeremy Rouse	CC BY-SA 4.0	Fixed a typo and added more clarity about the different.
Jul 5, 2022 at 18:53	history	answered	Jeremy Rouse	CC BY-SA 4.0