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I would like as many examples as possible (with explanation) of the object mentioned in the title (for which $p$ does it exists ? How do you construct those generically ? Is there a 'classification' ?)

In the case $p=3$, $ \mathbb{Q}_3[X]/(X^3-3)$ is an example. But I have to say I'm not particularly interested by the peculiarities of the $p=3$ case.

I guess lots of people will say 'class field theory'. I have to admit I know nothing about that, but if you can mention which combinations of theorem I have to use to understand the example(s), I'm of course happy !

user293657
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  • What do you mean by a non-cyclic extension? Usually, this would mean a Galois extension with non-cyclic Galois group, but there are no such cubic extensions. Do you perhaps mean "non-Galois" extension? – Mathmo123 Nov 26 '15 at 16:07
  • Sorry, the phrasing is unclear. I mean a non-normal cubic extension (so that the Galois group of the normal closure is $S_3$). So, yes, I mean non-Galois :-) – user293657 Nov 26 '15 at 16:13

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Actually, class field theory won't help you (at least not directly), since you're looking for non-abelian extensions!

Since you want examples, though, you should play with this wonderful database: https://math.la.asu.edu/~jj/localfields/

Away from $3$, your extension will be totally tamely ramified, and then it can always be written by taking the root of a uniformizer. At $3$, it's hard -- consult the tables!

hunter
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  • Indeed, that's a wonderful database. But still, I am also interested in understanding for example, how could I know by hand that there is a non-cyclic Galois extension of degree 6 of $\mathbb{Q}_5$ ? I guess now I could read the paper about the construction of the database, though ! – user293657 Nov 26 '15 at 16:26
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    @user293657 since 5 is prime to 3, the last sentence of my answer applies: take the cube root of a uniformizer (e.g. 5). – hunter Nov 28 '15 at 12:50
  • Argh, my question in comments is very silly (it just depends on whether there is a cubic root of unity in the residue field), and indeed, your last sentence applies to answer it (and even worse, I already knew it). Thanks very much for your patience ! – user293657 Nov 28 '15 at 15:20