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needed to point to ring of integers of a number field. The term used to redirect just to integers. I have split it off now with a minimum of content. Have to rush off now.
Fixed a mathematical typo and added a little more to the Idea section.
I made integer link to ring of integers in the Terminology section.
But we also have algebraic integer (written primarily by Todd) ,which seems to cover much the same ground. (It even has the boldface term ‘ring of integers’ in it, so the redirect probably should have been there all along.)
Oh, I didn’t see this, sorry. Should be merged.
Btw, thanks for fixing my text. I suppose I left out the condition that the coefficients are integra.
I gave both algebraic integer and ring of integers a brief Idea-section to help the reader with getting an idea of what’s going on before diving into the technical details. Please check if you agree and feel invited to edit as need be.
Do you still want to merge them?
I wouldn’t mind, but I don’t feel like doing it myself. If you want to, please do!
I probably will then, but not on this lousy phone!
I have added the definition to ring of integers for the case of a local non-archimedean field.
And I have expanded the Idea-sentence just a little and added two more simple examples in the Example-section.
I made a number of changes in ring of integers. Namely the entry had the format which sugested that the definitions in a number field and in a local nonarchimedean field are different and did not include other cases. I made it the standard way with easy general definition in an arbitrary commutative unital ring. The special cases are now under notation and properties as it should be.
I added a reference at ring of integers to the recently published paper Defining $\mathbb{Z}$ in $\mathbb{Q}$.
added to the list of examples:
The ring of intergers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$, called the ring of cyclotomic integers.
The statement made in ring of integers about the ring of integers of local fields as defined in the article aren’t correct in the characteristic p>0 case, are they? The element t in $F_p((t))$ is transcendental over $F_p$, so it can’t ever be an algebraic integer if you define it as the root of a polynomial with integer coefficients. So the ring of integers in $F_p((t))$ isn’t $F_p[[t]]$.
Thanks for the heads-up. Looks like I wrote this statement in revision 5 from 2014. I forget what I was thinking there and then, would need to remind myself.
But this must be textbook material and needs a citation in either case. What’s a decent reference to quote such basic examples from (once corrected)?
It’s contradictory for the characteristic zero case too, since most elements of $\mathbb{Z}_p$ aren’t algebraic integers.
I’m used to the general notion of “integral” being defined relative to a subring, rather than as an absolute notion. So $\mathbb{Z}_p \subseteq \mathbb{Q}_p$ is the ring of integers because we chose it to be rather than because of some general definition of “integer”.
added point to
for definitions etc.
Removed the line about Laurent polynomials for the moment. If nobody else does, I’ll come back to this later when I have the leisure.
I don’t think there was anything wrong with the Laurent series example, or that there is anything wrong with the $\mathbb{Q}_{p}$ example. These are non-archimedean local fields, and the definition given on the page in that case is being used. The definition concerning monic polynomials is only for algebraic number fields.
It might be best to give a definition which works in all cases, namely they are algebraic integers (in the non-archimedean local field sense in that case) in any completion.
Richard, thanks. This must be textbook material, do we have a canonical reference that could be quoted?
Cassels would be the canonical reference for local fields, but it’s years since I’ve looked at it. I’ll try to edit this page and a few surrounding ones if I get a chance at some point this week.
Thanks. I am scanning over this pdf copy of Cassels-Fröhlich, but haven’t spotted rings of integers yet…
I was thinking of the book ’Local fields’ by Cassels. Chapter 4 contains the definition for a field equipped with a non-archimedean valuation (a typical local field would be one obtained by completion of such a field with respect to the valuation), and Corollary 3 of Chapter 10 is the result I mentioned which relates the definition given on the page for algebraic number fields to the valuation-theoretic one (in particular, for any field, if one defines an algebraic integer to be one whose valuation is less than or equal to $1$ for any non-archimedean completion, one has a definition which works in all cases; one gets that there are ’no’ algebraic integers when the field does not admit any non-archimedean completions, such as $\mathbb{R}$ or $\mathbb{C}$, but that is OK).
It’s been a while, but IIRC $\mathbb{C} \cong \mathbb{C}_p$, so $\mathbb{C}$ does have nonarchimedean valuations. So any attempt at a purely algebraic definition is going to be unsatisfactory, because the cases of $\mathbb{C}$ and $\mathbb{C}_p$ “ought” to be different.
Does the definition give the thing you want in the nonarithmetic case? It sounds like the one you’re proposing would give that the integers in any function field $F(x)$ would be the ring of integers in $F$… but the subring of polynomials $F[x] \subseteq F(x)$ is the usual analog to $\mathbb{Z} \subseteq \mathbb{Q}$. (especially when $F = \mathbb{F}_p$)
Hehe, the isomorphism between $\mathbb{C}$ and $\mathbb{C}_{p}$ is an abomination, and if this were the only issue I wouldn’t be too concerned! I also wouldn’t be too worried about the function field case. However, it is probably best to just give the definition of ring of integers for a valued field as Cassels does (in complete generality), and then observe that for an algebraic number field, one can pick any non-archimedean valuation (one gets the same answer in all cases by the result I mentioned, and it agrees with the characterisation in terms of monic polynomials, etc).
I was probably too terse earlier, so I wanted to vote again that the general notion of “integer” be viewed as something depending on the context, rather than seeking an absolute definition gives you the “right” answer just from the algebraic object without the added choices. An example of what I mean by “choices” is choosing a set of valuations on a field, as I think #25 is alluding to, so that the associated notion of “integer” would be the intersection of the associated valuation rings.
Looking more at the history, I think my vote is to keep this page more to the original intention as the case of the algebraic integers in a number field? And maybe related facts such as the topological closure of $\mathcal{O}$ in one of its local fields is the associated complete discrete valuation ring.
Then for the more general theory, maybe fleshing out integral closure and/or Dedekind domain?
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