08/22/2012, 11:04 AM
considering ( associative ) extensions however , mathematicians would never extend q ( q >1 ) units to q+1 units.
and an odd prime amount of units is not an extention itself because prime has no real divisors.
see once again examples such as quaternions , multiplication tables etc ( galois is overkill )
if we have a number a1 x1 + a2 x2 + ... an xn and we add a new sign xn+1 then we need to define also x1*xn+1 , x2*xn+2 , ... and those need to be distinct , lin indep , and nonzero , hence we have a multiplication amount of elements -> not a prime and not n+1.
lagrange group theory basicly. ( i assume all know langrange theorem in group theory )
considering that and we want 3 signs , it seems we should stop trying to extend the normal + and - and start with 3 totally new signs !!
the addition remains the same but i propose the multiplication table is
that they act like the 3 roots of unity.
regards
tommy1729
and an odd prime amount of units is not an extention itself because prime has no real divisors.
see once again examples such as quaternions , multiplication tables etc ( galois is overkill )
if we have a number a1 x1 + a2 x2 + ... an xn and we add a new sign xn+1 then we need to define also x1*xn+1 , x2*xn+2 , ... and those need to be distinct , lin indep , and nonzero , hence we have a multiplication amount of elements -> not a prime and not n+1.
lagrange group theory basicly. ( i assume all know langrange theorem in group theory )
considering that and we want 3 signs , it seems we should stop trying to extend the normal + and - and start with 3 totally new signs !!
the addition remains the same but i propose the multiplication table is
that they act like the 3 roots of unity.
regards
tommy1729