later kummer was working towards a theory of cyclotomic fields. it had been realised (probably by several people) that failure to complete proofs in the general case of fermat's last theorem by factorisation using the roots of unity was for a very good reason: a failure of the fundamental theorem of arithmetic to hold in the rings generated by those roots of unity was a major obstacle. out of kummer's work for the first time came a study of the obstruction to the factorisation. we now recognise this as part of the ideal class group: in fact kummer had isolated the "p"-torsion in that group for the field of "p"-roots of unity, for any prime number "p", as the reason for the failure of the standard method of attack on the fermat problem (see regular prime).