Friday, January 15, 2016

E8 symmetry, harmony, and genetic code

Bee gave in Facebook a link to an article about a connection between icosahedron and E8 root system (see this). The article (I have seen an article about the same idea earlier but forgotten it!) is very interesting.

The article talks about a connection between icosahedron and E8 root system. Icosahedral group has 120 elements and its double covering 2× 120=240 elements. Remarkably, E8 root system has 240 roots. E8 Lie algebra is 248 complex-dimensional contains also the 8 commuting generators of Cartan algebra besides roots: it is essential that the fundamental representation of E8 co-incides with its adjoint representation. The double covering group of icosahedral group acts as the Weyl group E8. A further crucial point is that the Clifford algebra in dimension D=3 is 8-D.

One starts from the symmetries of 3-D icosahedron and ends up with 4-D root system F4 assignable to Lie group and also to E8 root system. E8 defines a lattice in 8-D Euclidian space: what is intriguing that dimensions 3,4, 8 fundamental in TGD emerge. To me this looks fascinating - the reasons will be explained below.

What I might have understood

I try to explain what I have possibly understood.

  1. The notion of root system is introduced. The negatives of roots are also roots but not other multiples. Root system is crystallographic if it allows a subset of roots (so called simple roots) such that all roots are expressible as combinations of these simple roots with coefficients having the same sign. Crystallographic root systems are special: they correspond to the fundamental weights of some Lie algebra. In this case the roots can be identified essentially as the quantum numbers of fundamental representations from which all other representations are obtained as tensor products. Root systems allow reflections as symmetries taking root system to itself. This symmetry group is known as Coxeter group and generalizes Weyl group. Both H3 and H4 are Coxeter groups but not Weyl groups.

  2. 3-D root systems known as Platonic roots systems (A3, B3, H3) assignable to the symmetries of tetrahedron, octahedron (or cube), and icosahedron (or dodecahedron) are constructed. The root systems consist of 3 suitably chosen unit vectors with square equal to 1 (square of reflection equals to one) and the Clifford algebra elements generated by them by standard Clifford algebra product. The resulting set has a structure of discrete group and is generated by reflections in hyper-planes defined by the roots just as Weyl group does. This group acts also on spinors and one obtains a double covering SU(2) of rotation group SO(3) and its discrete subgroups doubling the number of elements. Platonic symmetries correspond to the Coxeter groups for a "Platonic root system" generated by 3 unit vectors defining the basis of 3-D Clifford algebra. H3 is not associated with any Lie algebra but A3 and B3 are.

    Pinors (spinors) correspond to products of arbitrary/even number of Clifford algebra elements. They mean something else than usually a bein identified as elements of the Clifford algebra acting and being acted on from left or right by multiplication so that they always behave like spin 1/2 objects since only the left(right)-most spin is counted. The automorphisms involve both right and left multiplication reducing to SO(3) action and see the entire spin of the Clifford algebra element.

  3. The 3-D root systems (A3, B3, H3) are shown to allow an extension to 4-D root systems known as (D4, F4, H4) in terms of 3-D spinors. D4 and F4 are root systems of Lie algebras (see this). F4 corresponds to non-simply-laced Lie group related to octonions. H4 is not a root system of any Lie algebra.


  4. The observation that the dimension of Clifford algebra of 3-D space is 23=8 and thus allows imbedding of at most 8-D root system must have inspired the idea that it might be possible to construct the root system of E8 in 8-D Clifford algebra from 240 pinors of the double covering the 120 icosahedral reflections. Platonic solids would be behind all exceptional symmetry groups since E6 and E7 are subgroups of E8 and the construction should give their root systems also as low-dimensional root systems.

Mc Kay correspondence

The article explains also McKay correspondence stating that the finite subgroups of rotation group SU(2) correspond to simply laced affine algebras assignable with ADE Lie groups.

  1. One considers the irreducible representations of a finite subgroup of the rotation group. Let the number of non-trivial representations be m so that by counting also the trivial representation one has m+1 irreps altogether. In the Dynkin diagram of affine algebra of group with m-D Cartan algebra the trivial representation corresponds to the added node. One decomposes the tensor product of given irrep with the spin 2 representation into direct sum of irreps and constructs a diagram in which the node associated with the irrep is connected to those nodes for which corresponding representation appears in the direct sum. One can say that going between the connected nodes corresponds to forming a tensor product with the fundamental representation. It would be interesting to know what happens if one constructs analogous diagrams by considering finite subgroups of arbitrary Lie group and forming tensor products with the fundamental representation.

  2. The surprising outcome is that the resulting diagram corresponds to a Dynkin diagram of affine (Kac-Moody) algebra of ADE group with Cartan algebra, whose dimension is m. Cartan algebra elements correspond to tensor powers of fundamental representation: can one build any physical picture from this? For m= 6,7,8 one obtains E6, E7, E8. The result of the article implies that these 3 Lie-groups correspond to basis of 3 3-D unit identified as units of Clifford algebra: could this identification have some concrete meaning as preferred non-orthogonal 3-basis?

  3. McKay correspondence emerges also for inclusions of hyper-finite factors of type II1. The integer m characterizing the index of inclusion corresponds to the dimensions of Cartan algebra for ADE type Lie group. The inclusions of hyperfinite factors (HFFs) are characterized by integer m≥3 giving the dimension of Cartan algebra of ADE Lie groups (there are also C, F and G type Lie groups). m= 6,7,8 corresponds to exceptional groups E6, E7, E8 on one hand and to the discrete symmetry groups of tetrahedron, octahedron, icosahedron on the other hand acting as symmetries of corresponding 3-D non-crystallographic systems and not allowing interpretation as Weyl group of Lie group.

Connection with the model of harmony

These findings become really exciting from TGD point of view when one recalls that the model for bioharmony ( for 12-note harmonies central in classical music in general relies on icosahedral geometry. Bioharmonies would add something to the information content of the genetic code: DNA codons consisting of 3 letters A,T,C,G would correspond to 3-chords defining given harmony realized as dark photon 3-chords and maybe also in terms of ordinary audible 3-chords. This kind of harmonies would be roughly triplets of 3 basic harmonies and there would be 256 of them (the number depends on counting criteria). The harmonies could serve as correlates for moods and emotional states in very general sense: even biomolecules could have "moods". This new information should be seen in biology. For instance, different alleles of same gene are known to have different phenotypes: could they correspond to different harmonies? In epigenetics the harmonies could serve as a central notion and allow to realize the conjectured epigenetic code and histone code. Magnetic body and dark matter at them would be of course the essential additional element.

The inspiring observations are that icosahedron has 12 vertices - the number of notes in 12-note harmony and 20 faces- the number of amino-acids and that DNA codons consist of three letters - the notes of 3-chord.

  1. Given harmony would be defined by a particular representation of Pythagorean 12-note scale represented as self-non-intersecting path (Hamiltonian cycle) connecting the neighboring vertices of icosahedron and going through all 12 vertices. One assumes that neighboring vertices differ by one quint (frequency scaling by factor 3/2): quint scale indeed gives full octave when one projects to the basic octave. One obtains several realizations (in the sense of not being related by isometry of icosahedron) of 12-note scale. These realizations are characterized by symmetry groups mapping the chords of harmony to chords of the same harmony. These symmetry groups are subgroups of the icosahedral group: Z6, Z4, and two variants of Z2 (generated by rotation of π and by reflection) appear. Each Hamiltonian cycle defines a particular notion of harmony with allowed 3-chords identified by the 20 triangles of icosahedron.


  2. Pythagoras is trying to whisper me an unpleasant message: the quint cycle does not quite close! This is true. Musicologists have been suffering for two millenia of this problem. One must introduce 13th note differing only slightly from some note in the quint cycle. At geometrical level one must introduce tetrahedron besides icosahedron - only four notes and four chords and gluing along one side to icosahedron gives only one note more. One can keep tetrahedron also as disjoint from icosahedron as it turns out: this would give 4-note harmony with 4 chords something much simpler that 12- note harmony.


  3. The really astonishing discovery was that one can understand genetic code in this framework. First one takes three different types of 20-chord harmonies with group Z6, Z4, and Z2 defined by Hamiltonian cycles: this can be done in many different maners (there are 256 of them). One has 20+20+20 chords and one finds that they correspond nicely to 20+20+20=60 DNA codons: DNA codons coding for a given amino-acid correspond to the orbit of the triangle assigned with the amino-acid under the symmetry group of harmony in question.

    The problem is that there are 64 codons, not 60. The introduction of tetrahedron brings however 4 additional codons and gives 64 codons altogether. One can map the resulting 64 chord harmony to icosahedron with 20 triangles (aminoacids) and the degeneracies (number of DNA codons coding for given amino-acid in vertebrate code) come out correctly! Even the two additional troublesome amino-acids Pyl and Sec appearing in Nature and the presence of two variants of genetic code (relating to two kinds of Z2 subgroups) can be understood.

What could the interpretation of the icosahedral symmetry?

An open problem is the proper interpretation of the icosahedral symmetry.

  1. A reasonable looking guess would be that it quite concretely corresponds to a symmetry of some biomolecule: both icosahedral or dodecahedral geometry give rise to icosahedral symmetry. There are a lot of biomolecules with icosahedral symmetry, such as clathrate molecules at the axonal ends and viruses. Note that dodecahedral scale has 20 notes - this might make sense for Eastern harmonies - and 12 chords and there is only single dodecahedral Hamiltonian path found already by Hamilton and thus only single harmony. Duality between East and West might exist if there is mapping of icosahedral notes and to dodecahedral 5-chords and dodecahedral notes to icosahedral 3-chords and different notions of harmony are mapped to different notions of melody - whatever the latter might mean!).

  2. A more abstract approach tries to combine the above described pieces of wisdom together. The dynamical gauge group E8 (or Kac-Moody group) emerging for m=8 inclusion of HFFs is closely related to the inclusions for the fractal hierarchy of isomorphic sub-algebras of super-symplectic subalgebra. heff/h=n could label the sub-algebras: the conformal weights of sub-algebra are be n-multiples of those of the entire algebra.

    The integers ni resp. nf for included resp. including super conformal sub-algebra would be naturally related by nf= m× ni. m=8 would correspond to icosahedral inclusion and E8 would be the dynamical gauge group characterizing dark gauge degrees of freedom. The inclusion hierarchy would allow to realize all ADE groups as dynamical gauge groups or more plausibly, as Kac-Moody type symmetry groups associated with dark matter and characterizing the degrees of freedom allowed by finite measurement resolution.


  3. E8 as dynamical gauge group or Kac-Moody group would result from the super-symplectic group by dividing it with its subgroup representing degrees of freedom below measurement resolution. E8 could be the symmetry group of dark living matter. Bioharmonies as products of three fundamental harmonies could relate directly to the hierarchies of Planck constants and various generalized super-conformal symmetries of TGD! This convergence of totally different theory threads would be really nice!

Experimental indications for dynamical E8 symmetry

Lubos (thanks to Ulla for the link to the posting of Lubos) has written posting about experimental finding of E8 symmetry emerging near the quantum critical point of Ising chain at quantum criticality at zero temperature. Here is the abstract :

Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi–one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.

Phase transition leads from ferromagnetic to paramagnetic phase and spin excitations as pairs of kinks are replaced with spin flips (shortest possible pair of kinks and loss of the ferromagnetic order). In attempts to interpret the situation in TGD context, one must however remember that dynamical E8 is also predicted by standard physics so that one must be cautious in order to not draw too optimistic conclusions.

In TGD framework heff/h> 1 phases or phase transitions between them are associated with quantum criticality and it is encouraging that the system discussed is quantum critical and 1-dimensional.

  1. The large value of heff would be associated with dark magnetic body assignable to the magnetic fields accompanying the E8 "mesons". Zero temperature is not a prerequisite of quantum criticality in TGD framework.

  2. One should clarify what quantum criticality exactly means in TGD framework. In positive energy ontology the notion of state becomes fuzzy at criticality. For instance, it is difficult to assign the above described "mesons" with either ferromagnetic or paramagnetic phase since they are most naturally associated with the phase change. Hence Zero Energy Ontology (ZEO) might show its power in the description of (quantum) critical phase transitions.

    Quantum criticality could correspond to zero energy states for which the value of heff differs at the opposite boundaries of causal diamond (CD). Space-time surface between boundaries of CD would describe the transition classically. If so, then E8 "mesons" would be genuinely 4-D objects - "transitons" - allowing proper description only in ZEO. This could apply quite generally to the excitations associated with quantum criticality. Living matter is key example of quantum criticality and here "transitons" could be seen as building bricks of behavioral patterns. Maybe it makes sense to speak even about Bose-Einstein condensates of "transitons".

    The finding suggests that quantum criticality is associated with the transition increasing neff by factor m=8 or its reversal - maybe the standard value neff(i) =1. neff(f) =8 could correspond to the ferromagnetic phase having long range correlations. Could one could say that at the side of criticality (say the "lower" end of CD) the neff(f)=8 excitations are pure gauge excitations and thus "below measurement resolution" but become real at the other side of criticality (the "upper" end of CD)?

  3. The 8 "mesons" associated with spin excitations naturally correspond to the generators of the Cartan algebra of E8. If the "mesons" belong to the fundamental (= adjoint) representation of E8, one would expect 120+120 additional particles with non-vanishing E8 charges. Why only Cartan algebra? Is the reasons that Cartan algebra is in preferred role in the representations of Kac-Moody algebras in that charged Kac-Moody generators can be constructed from Cartan algebra generators by standard construction used also in string models. Could this explain why one expects only 8 "mesons". Are charged "mesons" labelled by the elements of double covering of icosahedral group more difficult to excite?

For the pdf version of the article see E8 symmetry, harmony, and genetic code.

For a summary of earlier postings see Links to the latest progress in TGD.

16 Comments:

At 6:55 AM, Anonymous Anonymous said...

Matti, what an entertaining post, very cool. MacKay. I remember something about moonshine and the MacKay-Thompson series. https://en.m.wikipedia.org/wiki/Monstrous_moonshine

The number 137 pops up again and again because it's the first time that the argument of zeta is not on the principal branch of the logarithm, also the numbers 3,4,8 etc also pop up after some relatively standard manipulations in my paper on hardy z function . I should have known Michael Berry is a cranky old man, he totally missed my point about about what I was suggesting and claimed "some of us discovered those graphs years ago" yet provided no evidence or references.. it's hard to find people to work with

--crow

 
At 1:42 PM, Blogger L. Edgar Otto said...

Matti, welcome back to the Glass Bead Game.

 
At 9:24 PM, Anonymous Matpitka6@gmail.com said...

Glass Bead Games with things like MacKay correspondence are dangerous (I might destroy my entire career;-) since my technical skills are so meagre. The only guide line is physical intuition and generalised common sense and I dare trust to them.

 
At 1:49 PM, Blogger Ulla said...

In spin ice with short-range interactions up to second neighbours, there is an intermediate critical phase separated from the paramagnetic and ordered phases by Kosterlitz–Thouless (KT) transitions. In dipolar spin ice, the intermediate phase has long-range order of staggered magnetic charges. The high- and low-temperature phase transitions are of the Ising and 3-state Potts universality classes, respectively. Freeze-out of defects in the charge order produce a very large spin correlation length in the intermediate phase. As a result of that, the lower-temperature transition appears to be of the KT type.

spin correlations decay with the distance algebraically rather than exponentially [logaritmic perpendicularly between different phases?] so spin is the slowest transition, always only 1/2 (fermion) and this should be the 'condensation phase' of the E8, of which most is non-commutative, and 'virtual'. What part of E8 is splitted and why? It must be the center, if I understand right, hence the creations of microBH and 'wormholes'are also possible. What role play the long correlation lengths? In Josephs simulations we saw it as a 'clock-function' of pulses.

You refer to gravitational couplings (and gravitational interaction?) but the use of constant G is a bit bothersome here in quantum level. Also it is normally eliminated by the hbar=c=G as 1, so it cannot be seen so well.

At the same time this describes different time scenarios and strong tensions.

 
At 1:50 PM, Blogger Ulla said...

Also link this to Wilson loops?

 
At 1:52 PM, Blogger Ulla said...

The link left out, sry.
http://rsta.royalsocietypublishing.org/content/370/1981/5718

 
At 8:27 PM, Anonymous Anonymous said...

Reminds me of the stochastic papers where I read a footnote that says "every occur ance of the constant C represents a different variable that changes with time"

 
At 10:30 PM, Anonymous Matpitka6@gmail.com said...


To Anonymous:

I do share the feelings of the authors of stochastic papers;-).

In the case of Planck constant it is however important that its spectrum is discrete and it reduces to group theory: the levels of fractal hierarchy of isomorphic subalgefbras of conformal algebra are labelled by integers n =h_eff/h. h van be taken to h=1 by a suitable choice of units.

Light velocity c can be reduced to c=1 by suitable choice of units when signal propagating with light-velocity is geometrize to the notion of light-like geodesic.

Newton's constant is geometrized to CP_2 size scale and one can take CP2 size as fundamental length unit.

Only Kahler coupling strength remains as dimensionless parameters, which cannot be eliminated by choice of units. It has a spectrum since it is inversely proportional to h_eff. The proposal is that this spectrum reduces essentially to the spectrum of zeros of zeta. Same would happen to other coupling strengths related to alpha_K.

 
At 10:34 PM, Anonymous Matpitka6@gmail.com said...

To Ulla:

The mentioning of Kosterlitz-Thouless forced me to Wikipedia to look summary about thermal phase transitions and this in turn to ask how TGD description could generalise the description of thermal and quantum phase transitions (quantum TGD as square root of thermodynamics). The reply however grew to entire posting so that I will not attach it to here.

 
At 10:52 AM, Anonymous Anonymous said...

http://arxiv.org/abs/1601.01797

This paper is excellent is astounding, I did not know that the H=xp Hamiltonian of Berry-Keating has interpretations in terms of general relativity and Rindler spacetimes. It goes into detail the interpretation of the "smooth Riemann zeros" that is, the zeros without the correction of the argument/phase of zeta.. and shows how the "missing spectral lines" interpretation of Connes is not entirely correct... the correct interpretation is that the argument provides a finite sized correct to the exact location of the zeros.. anyway, massive and massless fermions appear and they suggest that one could construct a system of reflecting moving mirrors that generates the zeros in its spectrum. Pretty cool

--crow

 
At 4:03 AM, Blogger Ulla said...

You say:
Newton's constant [G?] is geometrized to CP_2 size scale and one can take CP2 size as fundamental length unit.

Only Kahler coupling strength remains as dimensionless parameters, which cannot be eliminated by choice of units. It has a spectrum since it is inversely proportional to h_eff. The proposal is that this spectrum reduces essentially to the spectrum of zeros of zeta. Same would happen to other coupling strengths related to alpha_K.

Kähler coupling as c.c. or alpha? Also about the G seen as that, are there any links?
Forgive my stupidity...

 
At 4:42 AM, Anonymous Matpitka6@gmail.com said...


Kahler coupling strength requires some explanations.

*Kahler coupling strength alpha_K is the only coupling parameter of TGD and appears in Kahler action which defines the classical theory. alpha_K analogous to critical temperature: this if one accepts that TGD Universe is quantum critical.

*The question is whether alpha_K has just single critical value or large number of critical values. For instance, critical values could correspond to primes near to prime power of two: when p-adic length scale is scaled by suitable power of two, alpha_K changes. Coupling constant evolution would be discrete.

* This would allow to have non-trivial coupling constant evolution (albeit in discrete sense): this is forced by experimental facts. But alpha_K would also have trivial local coupling constant evolution lasting for few octaves for given prime satisfying this condition: alpha_K and other couplings would be piecewise constant functions of length scale.

* All radiative corrections would vanish and theory would be like N=4 SUSY in this respect: extremely simple. Also number theoretical considerations demand discrete coupling constant evolution.

*By symmetry arguments alpha_K analogous to weak U(1) coupling strength, which is not quite the same as fine structure constant alpha_em but near to it.

*The hypothesis that the values of 1/alpha_K interpreted in this manner correspond to zeros of zeta works so nicely that I tend to believe that this interpretation is correct. U(1) coupling constant evolution would reduce to number theory. Same would happen to other coupling strengths.

%%%%

I have considered many options. I had to give up the assumption that alpha_K at electron length scale corresponds to the value of alpha_em, fine structure constant. I have also been considering the possibility that alpha_K has just one value independent of p-adic length scale and considered also other formulas for alpha_K.

 
At 11:03 AM, Blogger Ulla said...

"For instance, critical values could correspond to primes near to prime power of two: when p-adic length scale is scaled by suitable power of two, alpha_K changes. Coupling constant evolution would be discrete."

As instance E7 has primes 3,5 and 7 inherent in its symmetry, so that 7 is bosonic, 3 and 5 fermionic (Pythagoras?)? But is the commutative part giving massivation/condensation also between 3 and 5 (compare DNA?), in fact a small part of the whole non-commutative Lie group. Actually like a 'surface' (compare Josephs simulation video), but that 'surface' can bulge out much? (Compare to arXiv: 1410.8447v1
http://irfu.cea.fr/Phocea/Vie_des_labos/Ast/ast.php?t=fait_marquant&id_ast=3533s)and 'sea quarks' necessary to have involved? Also twin primes interesting. This is why I asked about primes and/as thermodynamics.

Also, what is a hadron seems varying. Also 'annihilation' of whole hadron possible, and consequent 'popping up'. http://arxiv.org/abs/1406.7425 and the 'compactification' of sheets at higher N.

 
At 7:24 PM, Anonymous Matpitka6@gmail.com said...

The small primes 3,5,7, that you assign with E7 have nothing to do with p-adic primes. Symmetry groups and p-adic length scale hypothesis are totally unrelated.

%%%%

p-Adic primes of course appear in p-adic coupling constant evolution. This involves 3 conjectures: here I am little bit ashamed since I know that most conjectures made by human kind have been wrong. So: hereby I confess all my potential crimes against serious science;-):

Conjecture 1: The values of alpha_K analogous to critical temperature are conjectured to be labelled by p-adic primes near prime powers of two (p= about 2^k, k prime: zeros in increasing order for imaginary part<-->primes in increasing order. Every p-adic length scale identified in this manner corresponds to its own critical temperature. This allows to realize coupling constant evolution (physical fact) with vanishing loop corrections and thus also vanishing divergences). One could ask: why not all p-adic length scales, why not p-adic length scales for which k is integer?

Conjecture 2: Number theoretical universality (NTU) motivates the conjecture that primes labels zeros of zeta: exponents p^iy for imaginary parts y of zeros of zeta define roots of unity.

NTU derives from the vision about adelic physics. Various p-adic physics serve as cognitive representations for real physics. Strong form of holography making string like objects and partonic 2-surfaces basic objects realizes this vision at space-time level. Coupling constant evolution would be number theoretical and reflect the hierarchy of algebraic extensions of rationals at fundamental level.

Two conjectures are involved: this is of course dangerous!!

Prediction: The identification of spectrum of 1/alpha_K with zeros of zeta predicts an evolution which looks realistic and the prediction for value of alpha_U(1) is excellent at electron length scale.

Conjecture 3: The spectra of *all* gauge coupling constants in coupling constant evolution correspond to zeros of zeta(M(s), where M is some Mobius transformation s-->as+b/cs+d, a,b,c,d real, mapping which is holomorphic in upper half plane and maps its to itself.

Prediction: The evolution of 1/alpha_W, the weak coupling strength is obtained in this manner and the integers a,b,c,d are very simple: one of them is 137 (magic number!) and the values of Weinberg angle is correct and fine structure constant are correct at electron's p-adic length scale. At ultra high energies the predictions differ dramatically from those of standard model.

 
At 12:49 PM, Blogger Ulla said...

Why must the primes be p-adic? You also use Mersienne primes.
Note that these small primes are mostly 'virtual' roots?

Here a good monopole link http://dl.bsu.by/file.php/534/Lecture4.pdf

 
At 4:40 PM, Anonymous Matpitka6@gmail.com said...


When I talk about p-adic prime I talk about prime as prime characterizing p-adic number field.

The number of prime is very general and defined not only for rationals but also for their algebraic extensions. One of
them is complex rationals: in this case algebraic extension consists of complex integers m+in and Gaussian primes are primes for this number fields. There is infinity of other extensions and all of them have also p-adic variants. The number of number fields is
really huge.

Primes are well-defined also for quaterions and octonions.

 

Post a Comment

<< Home