Practice Questions for CSIR NET Ring Theory : Ring and Ideals - IV

31. Let \(\mathbf{F}\) be a field with \(5^{12}\) elements. What is the total number of proper subfields of \(\mathbf{F}\) ?

(a.) 3
(b.) 6
(c.) 4
(d.) 5

32. Which of the following statement is true.

(a.) \(\exists\) a field of order 36 but not order 49 .
(b.) \(\exists\) a field of order 36 .
(c.) \(\nexists\) field of order 36 and 49
(d.) \(\nexists\) field of order 36 .

33. Let \(\mathbb{Z}_{10}\) denote the ring of integers modulo 10. Then the number of ideals in \(\mathbb{Z}_{10}\) is

(a.) 2
(b.) 3
(c.) 4
(d.) 5

34. The number of subfields of a finite field of order \(3^{10}\) is equal to

(a.) 4
(b.) 5
(c.) 3
(d.) 10

35. If \(\mathrm{S}\) is a finite commutative ring with 1 then

(a.) Each prime ideal is a maximal ideal
(b.) \(\mathrm{S}\) may have a prime ideal which is not maximal
(c.) \(\mathrm{S}\) has no nontrivial maximal ideals
(d.) \(\mathrm{S}\) is a field

36. Suppose \((F,+, \cdot)\) is the finite field with 9 elements. Let \(G=(F,+)\) and \(H=(F \backslash\{0\}, \cdot)\) denote the underlying additive and multiplicative groups respectively. Then

(a.) \(G \cong(\mathbb{Z} / 3 \mathbb{Z}) \times(\mathbb{Z} / 3 \mathbb{Z})\)
(b.) \(G \cong(\mathbb{Z} / 9 \mathbb{Z})\)
(c.) \(H \cong(\mathbb{Z} / 2 \mathbb{Z}) \times(\mathbb{Z} / 2 \mathbb{Z}) \times(\mathbb{Z} / 2 \mathbb{Z})\)
(d.) \(G \cong(\mathbb{Z} / 3 \mathbb{Z}) \times(\mathbb{Z} / 3 \mathbb{Z})\) and \(H \cong(\mathbb{Z} / 8 \mathbb{Z})\)

37. The number of non-zero ideals of \(\frac{\mathbb{Z}}{100 \mathbb{Z}}\) is

(a.) 4
(b.) 8
(c.) 10
(d.) 6

38. If \(p\) is prime, and \(\mathbb{Z}_{p^{4}}\) denote the ring of integers modulo \(p^{4}\), then the number of maximal ideals in \(\mathbb{Z}_{p^{4}}\) is

(a.) 4
(b.) 2
(c.) 3
(d.) 1

39. For \(n \geq 1\), let \((\mathbb{Z} / n \mathbb{Z})^{*}\) be the group of units of \((\mathbb{Z} / n \mathbb{Z})\). Which of the following groups are cyclic?

(a.) \((\mathbb{Z} / 10 \mathbb{Z})^{*}\)
(b.) \(\left(\mathbb{Z} / 2^{3} \mathbb{Z}\right)^{*}\)
(c.) \((\mathbb{Z} / 100 \mathbb{Z})^{*}\)
(d.) \((\mathbb{Z} / 163 \mathbb{Z})^{*}\)

40. Let \(F_{125}\) be the field of 125 elements. The number of non-zero elements \(\alpha \in F_{125}\) such that \(\alpha^{5}=\alpha\) is ___.

(a.) 1
(b.) 5
(c.) 4
(d.) 25