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I see the ideals of an algebraic number field as lattices and prime ideals are the ones which you can't refine.

How can we form a picture of ideles and adeles?

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    $\begingroup$ please don't feel like not posting if you have an answer that depends on some (advanced) background. I will happily fill in background I lack. $\endgroup$
    – user58512
    Jan 30, 2013 at 21:06
  • $\begingroup$ I have provided an answer, is it what you are looking for? $\endgroup$
    – fretty
    Feb 4, 2013 at 12:14
  • $\begingroup$ @fretty ,not really but it was very illuminating thanks. I think my question was wrong. $\endgroup$
    – user58512
    Feb 4, 2013 at 12:56
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    $\begingroup$ Dear user, If you haven't seen it, you may want to read my answer on Tate's thesis. Regards, $\endgroup$
    – Matt E
    Mar 2, 2013 at 3:52

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I think the answer is that you really can't picture them in the same way as ideals.

The ring of adeles really consists of points described by infinitely many coordinates all belonging to different fields, the point is not to be able to picture them but to understand what they are supposed to represent. The reason ideals have such a pictorial representation is because number fields can be embedded into $\mathbb{C}$ and so we have geometry.

Considering two rings $R,S$, you know you can form their direct product $R\times S$. This consists of "coordinates" $(r,s)$ with $r\in R, s\in S$.

What the adeles belong to is an "infinite" version of this, the direct product $\Pi_{v} K_v$ of all different completions $K_v$ of the number field $K$. However it is not this full ring.

Already though we can see that there is no longer any way we can visualise these in the same way that we do for ideals, the components mostly lie in $p$-adic fields (apart from when $v$ is an archimedian place, in which case we do get elements inside $\mathbb{R}$ or $\mathbb{C}$ at those places). The best I can suggest is that maybe you can "picture" an adele via its archimedian components but this is a really crude way to view them since most of the important info is lost.

Ok so maybe I should say a little about the point of adeles. In fact I am going to do it through motivating the ideles which form the unit group of the ring of adeles.

By definition an idele of a number field $K$ is an element $(a_v)_v\in\Pi_{v} K_v^{\times}$ such that $a_v \in O_v^{\times}$ for all but finitely many non-archimedian $v$.

Why do we have this condition? Well the point is that it models the bahaviour seen by fractional ideals.

Given an idele $(a_v)_v$ of $K$ we know that each non-archimedean place $v$ has a prime ideal $\mathfrak{p}_v$ attatched to it (and vise versa in a one to one correspondence).

So we can create a fractional ideal from $(a_v)_v$ via $\Pi_{v}\mathfrak{p}_v^{\text{ord}_v(a_v)}$ where this product is over non-archimedean places $v$.

In fact the only ambiguity in this construction come from multiplication by units at each non-archimedean place...so we have proved that $\mathbb{I}_K / E_K \cong I_K$, where $\mathbb{I}_K$ are the ideles, $E_K$ is the subgroup where every component lies in $O_v^{\times}$ and $I_K$ is the group of fractional ideals of $K$.

We see that the "contained in $O_v^{\times}$ at all but finitely many places" condition is really capturing the fact that a fractional ideal only has finitely many primes with negative exponents in it's unique prime factorisation.

Adeles have similar structure and can be used quite a lot to explain stuff in number theory, such as Dirichlet's Unit Theorem and the finiteness of the Class number. They are more natural proofs than the ones you might have seen before.

Ideles have a natural history of being used in class field theory to explain how infinite abelian extensions work in a similar way to in which the ideal theory works for finite extensions.

In modern number theory the adeles are important when working with automorphic forms and algebraic groups.

Hope this answers your question.

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    $\begingroup$ fretty, nice explanation. is there any book or notes on class field theory which explains the points you mentioned; especially the mapping between adeles and ideals $\endgroup$
    – Math123
    Oct 8, 2014 at 9:29
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    $\begingroup$ Well the group of ideles is just the unit group of the ring of adeles. As for class field theory there are many good sources, the book by Childress is a good, down to earth source. I also have documents on my academia page: sheffield.academia.edu/DanielFretwell $\endgroup$
    – fretty
    Oct 8, 2014 at 21:40
  • $\begingroup$ @fretty, so there are infinitely many valuations $v$ on a number field $\endgroup$
    – MAS
    Dec 1, 2020 at 14:29
  • $\begingroup$ @fretty, also in para $9$, you said that each non-Archimedian places $v$ correspond to a prime ideal $\mathfrak{p}_v$. I think only $finite$ places $v$ has this property $\endgroup$
    – MAS
    Dec 1, 2020 at 14:38

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