Definition talk:Valued Field

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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 on 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 on 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)
BTW keep up the excellent work -- not many people have the patience to go through many pages just to change links. --prime mover (talk) 08:08, 1 November 2018 (EDT)

I surmise that the use of $\size {\,\cdot\,}$ on a field and $\norm {\,\cdot\,}$ on a vector space comes about for two reasons:
  • to suggest the similarity between the $\size {\,\cdot\,}$ on a field and the absolute value on $\R$ or $\C$ and the similarity between $\norm {\,\cdot\,}$ on a vector space and the Euclidean norm on $\R^n$ or $\C^n$. $\size {\,\cdot\,}$ involves multiplication, $\norm {\,\cdot\,}$ involves scalar multiplication.
  • to distinguish the norm $\size {\,\cdot\,}$ on a scalar field from the norm $\norm {\,\cdot\,}$ on the vector space. It would help distinguish scalars and vectors in equations.
While you can think of a field as a vector space its not natural and there seems to be little to be gained. You only think of $\R$ or $\C$ as one dimensional spaces in the context of an arbitrary n-dimensional space $\R^n$ or $\C^n$.
Most sources seem to refer to a norm on a field or ring as an absolute value. $\mathsf{Pr} \infty \mathsf{fWiki}$ does seem to be going against the grain here, but the distinction that I see $\mathsf{Pr} \infty \mathsf{fWiki}$ is making between absolute value and norm is that absolute value is related to the order on an integral domain. So long as this distinction is maintained all will be good. The downside of this distinction is that the complex modulus on $\C$ shouldn't be referred to as an absolute value and reference to $\C$ should be removed from the page on absolute values. The use of "Valued Field" for a field with a norm allows $\mathsf{Pr} \infty \mathsf{fWiki}$ to minimise the differences with sources that are cited.
Most sources seem to use $\size {\,\cdot\,}$ for a norm on a field or ring.
BTW I didn't come to $\mathsf{Pr} \infty \mathsf{fWiki}$ with these views. They have developed over the last few months. --Leigh.Samphier (talk) 09:58, 1 November 2018 (EDT)

Absolutely agree regarding the use of $\size {\,\cdot\,}$ in the contexts of $\R$ and $\C$. I only meant to suggest changing $\size {\,\cdot\,}$ to $\norm {\,\cdot\,}$ in the context of a general field. A decision needs to be made on which way do you want to go on $\mathsf{Pr} \infty \mathsf{fWiki}$. The use of $\size {\,\cdot\,}$ on a field is going to make it easier to be consistent with most sources that are cited. So I am now inclined to leave things as is and possibly change the definition of a Valued Field to use $\size {\,\cdot\,}$ rather than $\norm {\,\cdot\,}$ --Leigh.Samphier (talk) 09:58, 1 November 2018 (EDT)
I take on board what you say ... n the general context of the general valued field, then perhaps using $\norm {\,\cdot\,}$ is better than using $\size {\,\cdot\,}$. Then when we invoke the specific instance of the general valued field and call it $\R$ or $\C$, that's when we say: $\size {\,\cdot\,}$ is the "absolute value" or "complex modulus" -- and by Absolute Value is Norm and/or Complex Modulus is Norm, it is a norm and so "these objects are valued fields which go see."
Then we have the best of both worlds. --prime mover (talk) 11:15, 1 November 2018 (EDT)
35 years of maintaining software - you learn patience and the value of that patience. --Leigh.Samphier (talk) 09:58, 1 November 2018 (EDT)
Closer to 32 years myself, but then I started late. :-) --prime mover (talk) 11:15, 1 November 2018 (EDT)