(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . ,...

(2) Let Z/nZ be the set of n elements {0, 1, 2, . . . , n ? 1} with addition and multiplication modulo n. (a) Which element of Z/5Z is the additive identity? Which element is the multiplicative identity? For each nonzero element of Z/5Z, write out its multiplicative inverse. (b) Prove that Z/nZ is a field if and only if n is a prime number. [Hint: first work out why it’s not a field when n isn’t prime. Try some small examples, e.g. n = 4, n = 6.]

Solutions

Expert Solution

(a) The 0 (zero) element of Z/5Z is the additive identity.

The 1 (one) element of Z/5Z is the additive identity.

(b) If n is not prime, then the multiplicative inverse of each element does not exist. Fore example, let n = 6, then we have

So, we see that if n is not prime then at least (n-1)/2 and even number do not have their inverse. So, it can not be a field.

milcah answered 1 year ago

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