Talk:Finite extensions of local fields

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Equivalent conditions for an unramified extension

The equivalent conditions given for an unramified extension claim that L/K is unramified if and only if the residue extension is separable (see (ii)). I don't think this is true. For example, if the residue fields are finite (which they are for local fields, at least if you take the definition of a local field to be the completion of a global field at a non-archimedean place), then the residue extension is always separable, irrespective of whether L/K is unramified.

Secondly, while I can see the intending meaning behind the statement of (v), it is incorrect as stated. It currently says that L/K unramified is equivalent to the statement that every uniformizer of L is a uniformizer of K. This is not true, since a uniformizer of L need not even lie in K, even if the extension is unramified (just take a uniformizer of K and multiply it by a unit of L not in K). What it should say is that L/K unramified is equivalent to the statement that every uniformizer of K is a uniformizer of L (not the other way around). 68.149.186.143 (talk) 22:43, 25 March 2012 (UTC)[reply]

Thanks for correcting (v) in the "unramified" section. I've corrected (ii). 68.149.186.143 (talk) 17:34, 27 March 2012 (UTC)[reply]