I just want to know if there is an algorithm to find the primitive element of a given finite extension $F/k$ if the intermediate fields are given. I know how to approach it in particular examples picking linear combinations (the case $\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}$ for instance). If it does not exists, is there, then, an algorithm for the finite separable extension case or for extensions of $\mathbb{Q}$?

Thanks in advance.

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    $\begingroup$ Pick a random element. No, really. $\endgroup$ – Qiaochu Yuan May 14 '13 at 22:58
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    $\begingroup$ If you read the proof of the primitive element theorem you'll understand why Qiaochu's comment is accurate. $\endgroup$ – KCd May 14 '13 at 23:33
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    $\begingroup$ In Lisl Gaal's Classical Galois Theory there is a description of an algorithm for computing a primitive element. I don't have the book with me now but I can try to post it when I do unless someone else answers your question. $\endgroup$ – tghyde May 15 '13 at 17:32

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