Number Theory A Field is a set that has the same properties as the commutative ring plus a multiplicative inverse for all elements but 0 a*a-=1 mod n, for all a#0 Can chose whether to do an operation and then reduce modulo n, or reduce then do the operation, since reduction is a homomorphism from the ring of integers to the ring of integers modulo n a op b mod n= [a mod n op b mod n] mod n Where op can be +,-,orNumber Theory • A Field is a set that has the same properties as the commutative ring plus a multiplicative inverse for all elements but 0 – a*a-1 = 1 mod n, for all a  0 • Can chose whether to do an operation and then reduce modulo n, or reduce then do the operation, since reduction is a homomorphism from the ring of integers to the ring of integers modulo n – a op b mod n = [a mod n op b mod n] mod n – Where op can be +, -, or *