Even a hierarchy of infinite primes and corresponding number fields appears. At the first level of the hierarchy of infinite primes, the integer coefficients of a polynomial Q defining infinite prime have no common prime factors. P=Q hypothesis states that the polynomial P defining space-time surface is identical with a polynomial Q defining infinite prime at the first level of hierarchy.
However, finite fields, which appear naturally as approximations of p-dic number fields, have not yet gained the expected preferred status as atoms of the number theoretic Universe. Also additional constraints on polynomials P are suggested by physical intuition.
Here the notions of prime polynomial and concept of infinite prime come to rescue. Prime polynomial P with prime order n=p and integer coefficients smaller than p can be regarded as a polynomial in a finite field. The proposal is that all physically allowed polynomials are constructible as functional composites of prime polynomials satisfying P=Q condition.
One of the long standing mysteries of TGD is why preferred p-adic primes, characterizing elementary particles and even more general systems, satisfy the p-adic length scale hypothesis. The proposal is that p-adic primes correspond to ramified primes as factors of discriminant D of polynomial P(x). D=P condition reducing discriminant to a single prime is an attractive hypothesis for preferred ramified primes. M8-H duality suggests that the exponent exp(K) of Kähler function corresponds to a negative power D-k. Spin glass character of WCW suggests that the preferred ramified primes for, say prime polynomials of a given degree, and satisfying D=P, have an especially large degeneracy for certain ramified primes P, which are therefore of a special physical importance.
For a summary of earlier postings see Latest progress in TGD.