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Older comments[edit]

I don't understand the definition. What is the topology on adeles to make it a locally compact group? AFAICS the usual product topology is not locally compact in this case. EJ 14:51, 20 Aug 2004 (UTC)

Correct. But the restricted product topology is. The adeles as a whole are the union of more 'reasonable' subsets looking like product of p-adic integers over almost all p (compact), cartesian product with a finite product of p-adic numbers and the real numbers (locally compact). Every given adele is in such a 'reasonable' set.

Charles Matthews 16:24, 20 Aug 2004 (UTC)

Thanks for a reply. Notice that the restricted product page doesn't describe the topology either, which is why I asked in the first place. Just to make sure I understand you correctly: the res. prod. topology coincides with prod. topology on each 'reasonable' subset, thus a subset of the restricted product is open iff it is relatively open in each 'reasonable' subset; in other words, the res. prod. topology is inductively generated by prod. topologies on 'reasonable' subset. Is this correct?
EJ 11:16, 21 Aug 2004 (UTC)

Yes, that sounds right. Thanks for pointing out the deficiency of the RP page - should be fixed up. Charles Matthews 11:47, 21 Aug 2004 (UTC)

Idèle vs idele[edit]

I have reverted this change which changes idèle to idele throughout, and adds the alternative adèle, with an edit summary that fails to mention those changes. This is the sort of change that tends to produce strong reactions, and so should be discussed here per WP:BRD. For what it's worth, Neukirch (1999) uses the form idèle and adèle. Deltahedron (talk) 18:10, 17 April 2013 (UTC)[reply]

It was not an arbitrary change. You can see that there was no standard before my edit. I just standardized the spelling that has already been established both on Wikipedia (see also Adelic algebraic group) and in the mathematical literature (confer 1; 2, 3). Neukirch (1999) is obviously not the only English-language source on the subject. --Omnipaedista (talk) 02:03, 18 April 2013 (UTC)[reply]
It was a change made without discussion here and without a suitable edit summary. If the spelling has been standardised here on Wikipedia, please link to the discussion where that consensus was established. The spelling has clearly not been standardised in the mathematical literature, and if you think it has please cite an authoritative source. Google book searches do not constitute an authoritative source. Deltahedron (talk) 06:29, 18 April 2013 (UTC)[reply]
I should have indeed provided a suitable edit summary. As far as Wikipedia is concerned the "no accent" form has been "established" pragmatically. The original version of 'Adele ring' and the long-standing versions of the articles 'Adelic algebraic group', 'Idele class character' feature no accent. The addition of an accent here was made by a single user who him/herself was not even consistent about it. As far the mathematical literature is concerned I deem that there is no authoritative source available establishing a priori which is the standard, so we could for the moment rely on Google Book searches and ArXiv searches. In any case even if the Wikipedia community eventually chooses for some reason to employ the accented form of the word 'idele', it would be wise to do so consistently throughout Wikipedia, and also to employ the accented form of the word 'adele' to avoid confusion. --Omnipaedista (talk) 11:23, 18 April 2013 (UTC)[reply]

Source of the term "adele"[edit]

Oscar Goldman claimed to be the originator of the contraction. He was taking notes for a series of lectures by Chevalley, who referred to them as "additive ideles" at the time. According to Goldman, when Chevalley noticed the contraction Goldman was using, he liked it and adopted it. Goldman always insisted on the pronunciation "add-ell." — Preceding unsigned comment added by (talk) 19:05, 31 May 2013 (UTC)[reply]

This comment would be very much more valuable if you could provide a source for it. Colin McLarty (talk) 04:12, 9 June 2013 (UTC)[reply]

I personally heard Goldman's pronunciation of "add-ell." The story about taking notes from Chevalley is a personal communication from Goldman's student Robert Rubin. — Preceding unsigned comment added by (talk) 14:27, 12 June 2013 (UTC)[reply]


This page might have become bloated to the extent that it detracts from its usefulness. For instance, having the last 2/3rds of the introduction being introducing notation for the following lemmas makes for a bad article.

Perhaps it would be a good idea to cut down or move to the end the long lemma/definition sections, and have a reasonably concise introduction section defining the adeles and talking about their applications (i.e. a brief sketch of its appearance in class field theory/analogues to the ring of repartitions in geometry, etc)? (21 Dec, 2018)

OK, I've adressed this but I've not yet gone through the long section at the bottom to remove any repetition with the short section at the top. (7 Aug 2020)

To me, this article reads more like a textbook than a wiki article. Unfortunately I not experienced in this field, but I think the article should certainly have less high-level, textbooky explanation of the issue than it does. The current article might be good for someone trying to research the topic, but not for someone trying to get a base-level explaination of an unfamiliar term. Perhaps the article in its current form might be better suited on Wikibooks or Wikiversity? Ihvt (talk) 08:43, 22 December 2022 (UTC)[reply]

I agree the length seems an issue and the style is also textbooky in a bad way. One simple solution is to move the idele group stuff to a separate article. —- Taku (talk) 15:27, 23 December 2022 (UTC)[reply]

Comment on valuations[edit]

In general the word "valuation" is sometimes used to mean the same thing as absolute value. However in the section on valuations, valuations are being referred to in their additive sense, i.e. with . They are then being directly contrasted with absolute values.

This section implies that there are archimedean valuations and non-archimedean valuations. The section makes no reference to a definition of additive valuation which includes archimedean cases, instead linking to the wikipedia page on valuations, which excludes archimedean cases. To my knowledge, there is no standardized general abstract additive definition of valuations which includes archimedean cases. In my opinion this warrants a clarification of the section.

Additionally, the section seems to conflate infinite places of with archimedean places of ; these can only be identified when is a number field, as infinite places of function fields are non-archimedean.

The section also has a few grammatical errors. 2600:6C4E:D7F:43CC:693D:F6AA:F5ED:5254 (talk) 05:17, 8 June 2024 (UTC)[reply]