Let's assume we are dealing with a GF($2^8$) - not necessarily related to the AES algorithm. It seems that the steps to perform multiplication of two elements of the field are:
1) multiply the polynomials as usual
2) divide the result with an irreducible polynomial from the field
How do I know which one of the irreducible polynomials to use? The necklace formula suggest that there are many. Is there a difference when using different irreducible polynomials? If so, do we then have many possible multiplication operators for the field? If not, why is there no difference?
Also, it seems that the multiplicative inverse is defined only once - is it somehow tied to a specific irreducible polynomial?
And a last, possibly related question: when I see field defined by a irreducible polynomial - is that perhaps tied to choosing a particular multiplication function?