Definition talk:Valued Field

Removing Absolute Value on a field
There are a number of links (~48) to Definition:Absolute Value (Field) which are now redirected to Definition:Norm/Division Ring. The links are generally of the form:
 * Let $\mathbb K$ be a field with absolute value $\left\vert{\cdot}\right\vert$.

Since Absolute Value (Field) has not really been defined I think these links should be removed. I'd like to change these instances to something like:
 * Let $\struct {\mathbb K, \left\vert{\,\cdot\,}\right\vert }$ be a valued field.

Similarly for Definition:Abstract Absolute Value the links are redirected to Definition:Absolute Value/Norm Theory where it is suggested that the page be merged with Definition:Norm/Division Ring. Again the links are of the form:
 * Let $k$ be a field with absolute value $\left\vert{\cdot}\right\vert$

which I would suggest replacing with:
 * Let $\struct {k, \left\vert{\,\cdot\,}\right\vert }$ be a valued field.

Is there any reason not to do this?

I know that $\left\vert{\,\cdot\,}\right\vert$ should also be replaced with $\norm {\,\cdot\,}$, but that will require many other changes in the pages and is not something I'll get through in an evening. --Leigh.Samphier (talk) 07:31, 1 November 2018 (EDT)


 * My preferred option would be


 * Let $\struct {\mathbb K, \size {\,\cdot\,} }$ be a valued field.


 * It keeps it simple and consistent.


 * I don't properly understand the difference in nuance between $\size {\,\cdot\,}$ and $\norm {\,\cdot\,}$, apart from having a vague idea that $\size {\,\cdot\,}$ is an instance of a $\norm {\,\cdot\,}$ which is used in certain contexts (e.g. as the absolute value function on $\R$, or as the complex modulus on $\C$.


 * But what I want to avoid is confusing students with a "basic" understanding of a metric space or the use of $\size {\,\cdot\,}$ in the contexts of $\R$ and $\C$, without needing to know anything about "norm theory". To the undergraduate who has not touched topology, all they need is $\size {\,\cdot\,}$. --prime mover (talk) 07:43, 1 November 2018 (EDT)