(a)Show that S = {a+b √ 5 | a, b ∈ Q} is a subring of the realnumbers (with the usual + and × of real numbers). Explain why S isa field.
(b) Prove that if r is an element of a ring R and r 3 = 0, then1 − r is a unit in R.
(c) Write down all the nilpotent elements of Z24, stating theindex of nilpotence in each case. Verify the statement in part (b)holds in Z24.
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