Which of the following sets are rings with respect to the usual operations of ad

Question

Which of the following sets are rings with respect to the usual operations of addition and multiplication? If the set is a ring, is it also a field? (a) 7Z (b) Z_18 (c) Q(squareroot 2) = {a + b squareroot 2: a, b elementof Q} (d) Q(squareroot 2, squareroot 3) = {a + b squareroot 2 + c squareroot 3 + d squareroot 6: a, b, c, d elementof Q} (e) Z[squareroot 3] = {a + b squareroot 3: a, b elementof Z} (f) R = {a + b 3 squareroot 3: a, b elementof Q} (g) Z[i] = {a + bi: a, b elementof Z and i^2 = -1} (h) Q(3 squareroot 3) = {a + b 3 squareroot 3 + c 3 squarerooot 9: a, b, c elementof Q}

Explanation / Answer

a) It is a ring but not a field due to the absence of inverse elements.

b) Z18 is not a field since 18 is not a prime Zn is a field iff n is a prime no. but it is also a ring.

c) Since Q(2(1/2)) is a field extension of field Q hence it is a field hence ring also.

d) from the same logic given in c d is also a field.

e) Z[3(1/2)] is an integral domain since Z[root d] is always an integral domain but it is not a field since 2+3(1/2) does not has an inverse in Z[root 3] because 1/7 doesnt belong to Z.

f) not a ring since it is not closed under multiplication.

g) ring of gaussian integers, but not a field due to the absence of inverse same as in part e.

h) It is also a field extension of Q so a field and hence a ring.

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