So, I've been told to research a little about GAP functions related with fields, this far I can get the "standard" representation of any field with GF(p) or GF(p^r) and a representation of GF(p^r) given the polynomial used to extend GF(p), pol by using AlgrebraicExtension(GF(p), pol). Now I was wondering is there is any way of generating a field if you were given both tables.
I've tried some things using Magma's and Ring's commands but I haven't been able to get anything this far.
Thanking you in advance!
While the general system setup would not preclude you from implementing such functionality (you'd likely end up with needing 500-1000 lines of code to implement everything needed), there isn't any existing function that would create a field from a multiplication table.
A sneaky (in letter but not in spirit) way of answering (I presume your tables are finite) would be to take
GF(Length(multiplicationtable))if the table is of size $\le 2^{16}$. (but that would not give you a correspondence between table indices and field elements).