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This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in an extension, depending on a chosen integer that generates the ring of integers.

But my question is,

"How does one show in general that for finite extensions $L / K$, there exists an algebraic integer $\alpha \in L$, so that $K[\alpha] = \mathcal O_L$? "

A related question is,

If $L$ is the compositum of subfields $K_1$ and $K_2$ over $K$, then can $\mathcal O_L$ be described in terms of $\mathcal O_{K_1}$ and $\mathcal O_{K_2}$? You may assume any additional conditions or even special cases like quadratic and quartic extensions.

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One doesn't show that in general because it's false in general (e.g. take $K = \mathbb{Q}, \alpha = \sqrt{-3}$.) – Qiaochu Yuan Feb 23 at 1:23
Qiaochu, I have edited the question. – Abhishek Parab Feb 23 at 1:27
That is also false in general. (I don't have a counterexample ready off the top of my head though.) – Qiaochu Yuan Feb 23 at 1:27
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Example 8.1.6 in Stein's A Brief Introduction to Classical and Adelic Number Theory gives a counterexample: the ring of integers in $\mathbb{Q}(\alpha)$ where $\alpha^3 + \alpha^2 - 2 \alpha + 8 = 0$ is not of the form $\mathbb{Z}[\beta]$ for any $\beta$. – Qiaochu Yuan Feb 23 at 1:30
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Dear Abhishek, you may find a partial answer to your second question in my answer here. – Bruno Feb 23 at 1:31
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