MATH21112 Rings and Fields
Example Sheet 4
Fields, Nilpotents and Idempotents
1. Show that Q[i] = {a + bi j a, b ∈ Q} (where i2 = -1) is a field.
2. Let R be the polynomial ring Z8 [X]. Show that the polynomial 1 + 2X is invertible in R.
(Hint: consider powers of 1 + 2X.)
3. Let R be a commutative ring. Prove that if a and b are nilpotent elements of R, then a + b is nilpotent.
4. Suppose that R is a ring such that a2 = a for every a ∈ R. Show that a = -a for all a ∈ R.
Show that R is commutative. (Hint: consider (a + b)2 .)
5. Prove that if R is a domain then there are no nilpotent elements other than 0 and no idempotent elements other than 0 and 1.
6. Find all idempotent and all nilpotent elements in the ring Z6 × Z12 .
7. Let R be a finite integral domain. Prove that R is a field.
(Hint: use a similar argument to that used in Lemma 2.12 where we showed that Zp is an integral domain and a field when p is prime.)