abstract algebra only problem 5, 6 and 8.

(2) Let P be a set of prime numbers, not necessarily nite. Consider the set S of all rational numbers whose denominators in the reduced forms only have prime divisors from P. (a) Show that S is an integral domain with respect to the sum and product in Q; (b) Describe the elements in S, which have multiplicative inverses (within S); (c) Describe the sets P for which 8 is a eld. (3) Give an example of a commutative unitary ring R such that (a + b)2 = a,2 + b2 for all a, b E R. (4) Give an example of a commutative unitary ring R such that (a + b)3 = a3 + b3 for all a, b E R. (5) Show that the following subset is eld w.r.t. the sum and product in R: (xx/5) = {a+b\\/ | a,beQ} CR. (6) Show that the set of continuous functions f : R > R is a commutative ring with 1 with respect to the usual (i.e., point-wise) addition and product. (7) Show that the set of polynomial functions f : R > R is a commutative ring with 1 with respect to the usual (i.e., point-wise) addition and product. (8) Which of the rings in Problems (5) and (6) is an integral domain? Which of them is a eld