Definition talk:Field Norm

From ProofWiki
Jump to navigation Jump to search

I removed the (false) item in the "Also See" section, which was:

It was added by Lord Farin, who in a later edit realized its falseness. Besides considering to rename it, adding something like "Not to be confused with Definition:Norm on Division Ring" or "For the definition of a norm on a field; see Definition:Norm on Division Ring" seems useful either way. --barto (talk) 12:03, 13 July 2017 (EDT)

More immediately interesting to my mind (yes I know this is not the exact same page) is how Definition:Automorphism Group of Field Extension seems to have the same definition as Definition:Galois Group.
It would be delightful if all this area could be tidied up and defined rigorously and uniquely. --prime mover (talk) 12:11, 13 July 2017 (EDT)
Incidentally, all this "perhaps" renaming, I repeat: what do your source works call it?
Are these definitions definitely the same thing? It appears that one is on a Galois extension, and one is not, in which case they are not the same thing at all and cannot be passed off as Definition 1 and Definition 2 of the same thing. --prime mover (talk) 12:21, 13 July 2017 (EDT)
The renaming was a suggestion of Lord Farin. I'm fine with this name. If any more specific name than just "norm" or "norm map" is used, it is "field norm". If this is found too ambiguous, one can name this page "Norm of Element in Field Extension". --barto (talk) 12:28, 13 July 2017 (EDT)
>>Are these definitions definitely the same thing? Yes. Definition 2 can only be given for Galois extensions. The question is: Suppose a definition only works for a subset of the range of objects it can be applied to, but coincides with a more general definition. Do we
1. Create a new section, new numbering (as has been done in most cases)?
2. Continue the numbering but add additional conditions to the definitions?
3. Choose between 1. and 2. depending on how serious the additional conditions are?
For example, the definitions of order of an entire function work respectively for: all functions but constant ones, all functions but the zero function, all functions. Is it worth creating a new section for the order of a constant function? (This is an extreme example. I'm just giving an overview of the possible situations, so we can decide about a policy.) --barto (talk) 13:02, 13 July 2017 (EDT)
A plus of 1 is that it makes immediately clear to the reader that we're imposing additional conditions. A disadvantage is that it hides the fact that the definitions do coincide, and that they have to be proven to be equivalent. --barto (talk) 13:05, 13 July 2017 (EDT)
If they are the same thing, just in a different context, then the one definition does for both. If, in the second, more restrictive context (as in this one, of Galois extensions), then the recommendation is that the characteristic given here that makes it a definition should be established as a proof, e.g. "Characteristic of 'Field Norm' (or whatever we call it) of Galois Extension", and when this characteristic is required in any further proof that requires this characteristic, it should be invoked, as:
"From [[Characteristic of 'Field Norm' (or whatever we call it) of Galois Extension]]:"
$N_{L / K} \left({\alpha}\right) = \displaystyle \prod_{\sigma \mathop \in \operatorname{Gal} \left({L / K}\right)} \sigma \left({\alpha}\right)$
... with appropriate explanatory words so as to reduce its gnomicity quotient. --prime mover (talk) 14:15, 13 July 2017 (EDT)
Yes, makes sense. But what if both definitions are seen in literature? --barto (talk) 14:20, 13 July 2017 (EDT)