Two instances of the inverse Galois problem, which are especially interesting in TGD, are following:
Q1: Can a given finite group appear as Galois group over Q? The answer is not known.
Q2: Can a given finite group G appear as a Galois group over some EQ? Answer to Q2 is positive as will be found and the extensions for a given G can be explicitly constructed.
The TGD based formulation based on M8-H duality in which space-time surface in complexified M8 are coded by polynomials with rational coefficients involves the following open question.
Q: Can one allow only polynomials with coefficients in Q or should one allow also coefficients in EQs?
The idea allowing to answer this question is the requirement that TGD adelic physics is able to represent all finite groups as Galois groups of Q or some EQ acting physical symmetry group.
If the answer to Q1 is positive, it is enough to have polynomials with coefficients in Q. It not, then also EQs are needed as coefficient fields for polynomials to get all Galois groups. The first option would be the more elegant one.
The inverse problem is highly interesting from the perspective of TGD. Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. The TGD based model of the genetic code involves in an essential manner the groups A5 (icosahedron), which is the smallest simple and non-commutative group, and A4 (tetrahedron). The identification of these groups as Galois groups leads to a more precise view about genetic code and answers to a key open question of the model in its recent form.
For a summary of earlier postings see Latest progress in TGD.