The intriguing general result of class field theory) something extremely abstract for physicist's brain  is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this).
Does this mean that it is not absolutely necessary to introduce padic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also padic integers and numbers are. What is intriguing that pdic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2adic units consisting of bit sequences which can have literally infinite nonvanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and padic norm approaching to zero expresses the reduction of the insignificance.
1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions?
Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.
 Could real physics and various padic physics on one hand, and number theoretic physics based on maximal Abelian extension of rational octonions and quaternions on one hand, define equivalent formulations of physics?
 Besides various padic physics all classical number fields (reals, complex numbers, quaternions, and octonions) are central in the number theoretical vision about TGD. The technical problem is that padic quaternions and octonions exist only as a ring unless one poses some additional conditions. Is it possible to pose such conditions so that one could define what might be called quaternionic and octonionic adeles and ideles?
It will be found that this is the case: padic quaternions/octonions would be products of rational quaternions/octonions with a padic unit. This definition applies also to algebraic extensions of rationals and makes it possible to define the notion of derivative for corresponding adeles. Furthermore, the rational quaternions define noncommutative automorphisms of quaternions and rational octonions at least formally define a nonassociative analog of group of octonionic automorphisms (see this).
 I have already earlier considered the idea about Galois group as the ultimate symmetry group of physics. The representations of Galois group of maximal Abelian extension (or even that for algebraic numbers) would define the quantum states. The representation space could be group algebra of the Galois group and in Abelian case equivalently the group algebra of ideles or adeles. One would have wave functions in the space of ideles.
The Galois group of maximal Abelian extension would be the Cartan subgroup of the absolute Galois group of algebraic numbers associated with given extension of rationals and it would be natural to classify the quantum states by the corresponding quantum numbers (number theoretic observables).
If octonionic and quaternionic (associative) adeles make sense, the associativity condition would reduce the analogs of wave functions to those at 4dimensional associative submanifolds of octonionic adeles identifable as spacetime surfaces so that also spacetime physics in various number fields would result as representations of Galois group in the maximal Abelian Galois group of rational octonions/quaternions. TGD would reduce to classical number theory!
 Absolute Galois group is the Galois group of the maximal algebraic extension and as such a poorly defined concept. One can however consider the hierarchy of all finitedimensional algebraic extensions (including nonAbelian ones) and maximal Abelian extensions associated with these and obtain in this manner a hierarchy of physics defined as representations of these Galois groups homomorphic with the corresponding idele groups.
 In this approach the symmetries of the theory would have automatically adelic representations and one might hope about connection with Langlands program.
2. Adelic variant of spacetime dynamics and spinorial dynamics?
As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the spacetime dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).
 Could one think that adeles or ideles could extend reals in the formulation of the theory: note that reals are included as Cartesian factor to adeles. Could one speak about adelic or even idelic spacetime surfaces endowed with adelic or idelic coordinates? Could one formulate variational principle in terms of adeles so that exponent of action would be product of actions exponents associated with various factors with Neper number replaced by p for Z_{p}. The minimal interpretation would be that in adelic picture one collects under the same umbrella real physics and various padic physics.
 Number theoretic vision suggests that 4:th/8:th Cartesian powers of adeles have interpretation as adelic variants of quaternions/ octonions. If so, one can ask whether adelic quaternions and octonions could have some number theretical meaning. Note that adelic quaternions and octonions are not number fields without additional assumptions since the moduli squared for a padic analog of quaternion and octonion can vanish so that the inverse fails to exist.
If one can pose a condition guaranteing the existence of inverse, one could define the multiplicative group of ideles for quaternions. For octonions one would obtain nonassociative analog of the multiplicative group. If this kind of structures exist then fourdimensional associative/coassociative submanifolds in the space of nonassociative ideles define associative/coassociative ideles and one would end up with ideles formed by associative and
coassociative spacetime surfaces.
 What about equations for spacetime surfaces. Do field equations reduce to separate field equations for each factor? Can one pose as an additional condition the constraint that padic surfaces provide in some sense cognitive representations of real spacetime surfaces: this idea is formulated more precisely in terms of padic manifold concept (see this). Or is this correspondence an outcome of evolution?
Physical intuition would suggest that in most padic factors spacetime surface corresponds to a point, or at least to a vacuum extremal. One can consider also the possibility that same algebraic equation describes the surface in various factors of the adele. Could this hold true in the intersection of real and padic worlds for which rationals appear in the polynomials defining the preferred extremals.
 To define field equations one must have the notion of derivative. Derivative is an operation involving division and can be tricky since adeles are not number field. If one can guarantee that the padic variants of octonions and quaternions are number fields, there are good hopes about welldefined derivative. Derivative as limiting value df/dx= lim ( f(x+dx)f(x))/dx for a function decomposing to Cartesian product of real function f(x) and padic valued functions f_{p}(x_{p}) would require that f_{p}(x) is nonconstant only for a finite number of primes: this is in accordance with the physical picture that only finite number of padic primes are active and define "cognitive representations" of real spacetime surface. The second condition is that dx is proportional to product dx × ∏ dx_{p} of differentials dx and dx_{p}, which are rational numbers. dx goes to xero as a real number but not padically for any of the primes involved. dx_{p} in turn goes to zero padically only for Q_{p}.
 The idea about rationals as points commont to all number fields is central in number theoretical vision. This vision is realized for adeles in the minimal sense that the action of rationals is welldefined in all Cartesian factors of the adeles. Number theoretical vision allows also to talk about common rational points of real and various padic spacetime surfaces in preferred coordinate choices made possible by symmetries of the imbedding space, and one ends up to the vision about life as something residing in the intersection of real and padic number fields. It is not clear whether and how adeles could allow to formulate this idea.
 For adelic variants of imbedding space spinors Cartesian product of real and padc variants of imbedding spaces is mapped to their tensor product. This gives justification for the physical vision that various padic physics appear as tensor factors. Does this mean that the generalized induced spinors are infinite tensor products of real and various padic spinors and Clifford algebra generated by induced gamma matrices is obtained by tensor product construction? Does the generalization of massless Dirac equation reduce to a sum of d'Alembertians for the factors? Does each of them annihilate the appropriate spinor? If only finite number of Cartesian factors corresponds to a spacetime surface which is not vacuum extremal vanishing induced Kähler form, Kähler Dirac equation is nontrivial only in finite number of adelic factors.
3. Objections
The basic idea is that appropriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.
 The first objection is that padic quaternions and octonions do not make sense since padic variants of quaternions and octonions do not exist in general. The reason is that the padic norm squared ∑ x_{i}^{2} for padic variant of quaternion, octonion, or even complex number can vanish so that its inverse does not exist.
 Second objection is that automorphisms of the ring of quaternions (octonions) in the maximal Abelian extension are products of transformations of the subgroup of SO(3) (G_{2}) represented by matrices with elements in the extension and in the Galois group of the extension itself. Ideles separate out as 1dimensional Cartesian factor from this group so that one does not obtain 4field (8fold) Cartesian power of this Galois group.
If the padic variants of quaternions/octonions are be rational quaternions/octonions multiplied by padic number, these objections can be circumvented.
 This condition indeed allows to construct the inverse of padic quaternion/octonion as a product of inverses for rational quaternion/octonion and padic number! The reason is that the solutions to ∑ x_{i}^{2}=0 involve always padic numbers with an infinite number of pinary digits  at least one and the identification excludes this possibility.
 This restriction would give a rather precise content for the idea of rational physics since all padic spacetime surfaces would have a rational backbone in welldefined sense.
 One can interpret also the quaternionicity/octonionicity in terms of Galois group. The 7dimensional nonassociative counterparts for octonionic automorphisms act as transformations x→ gxg^{1}. Therefore octonions represent this group like structure and the padic octonions would have interpretation as combination of octonionic automorphisms with those of rationals.
Adelic variants of of octonions would represent a generalization of these transformations so that they would act in all number fields. Quaternionic 4surfaces would define associative local subgroups of this grouplike structure. Thus a generalization of symmetry concept reducing for solutions of field equations to the standard one would allow to realize the vision about the reduction of physics to number theory.
See the new chapter Unified Number Theoretical Vision or the
article with the same title.
