Abstract Algebra: Theory and Applications · Problem 16.7.1

Rings — Thomas Judson

Mathematics — Abstract Algebra · rings

Problem

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) 7Z7\mathbb{Z} (b) Z18\mathbb{Z}_{18} (c) Q(2)={a+b2:a,b∈Q}\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} : a, b \in \mathbb{Q}\} (d) Q(2,3)={a+b2+c3+d6:a,b,c,d∈Q}\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \{a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6} : a, b, c, d \in \mathbb{Q}\} (e) Z[3]={a+b3:a,b∈Z}\mathbb{Z}[\sqrt{3}] = \{a + b\sqrt{3} : a, b \in \mathbb{Z}\} (f) R={a+b33:a,b∈Q}R = \{a + b\sqrt[3]{3} : a, b \in \mathbb{Q}\} (g) Z[i]={a+bi:a,b∈Z and i2=−1}\mathbb{Z}[i] = \{a + bi : a, b \in \mathbb{Z} \text{ and } i^2 = -1\} (h) Q(33)={a+b33+c93:a,b,c∈Q}\mathbb{Q}(\sqrt[3]{3}) = \{a + b\sqrt[3]{3} + c\sqrt[3]{9} : a, b, c \in \mathbb{Q}\} Hint. (a) 7Z7\mathbb{Z} is a ring but not a field; (c) Q(2)\mathbb{Q}(\sqrt{2}) is a field; (f) RR is not a ring.

Browse the problem library