Practice Questions for CSIR NET Ring Theory : UFD, PID and ED II

11. Consider \(\mathbb{Z}_{5}\) as the field modulo 5 and let \(f(x)=x^{5}+4 x^{4}+4 x^{3}+4 x^{2}+x+1\). Then the zeros of \(f(x)\) over \(\mathbb{Z}_{5}\) are 1 and 3, with respective multiplicities

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

12. Let \(\mathrm{R}\) be the polynomial ring \(\mathbb{Z}_{2}[x]\) and write the elements of \(\mathbb{Z}_{2}\) as \(\{0,1\}\). Let \(\langle f(x)\rangle\) denote the ideal generated by the element \(f\) \((x) \in \mathrm{R}\). If \(f(x)=x^{2}+x+1\), then the quotient ring \(\mathrm{R} /\langle f(x)\rangle\) is

(a.) a ring but not an integral domain.
(b.) an integral domain but not a field.
(c.) a finite field of order 4.
(d.) an infinite field.

13. Let \(\mathbb{R}[x]\) be the polynomial ring in \(x\) with real coefficients, and let \(I=\left\langle x^{2}+1\right\rangle\) be the ideal generated by the polynomial \(x^{2}+1\) in \(R[x]\). Then

(a.) \(I\) is a maximal ideal
(b.) \(I\) is a prime ideal but NOT a maximal ideal
(c.) \(I\) is NOT a prime ideal
(d.) \(\mathrm{R}[x] / \mathrm{I}\) has zero divisors

14. Let \(F_{4}, F_{8}\), and \(F_{16}\) be finite fields of 4, 8, and 16 elements respectively. Then,

(a.) \(F_{4}\) is isomorphic to a subfield of \(F_{8}\)
(b.) \(F_{8}\) is isomorphic to a subfield of \(F_{16}\)
(c.) \(F_{4}\) is isomorphic to a subfield of \(F_{16}\)
(d.) None of the above

15. Let PID, ED, UFD denote the set of all principal ideal domains, Euclidean domains, unique factorization domains, respectively. Then

(a.) \(U F D \subset E D \subset P I D\)
(b.) \(P I D \subset E D \subset U F D\)
(c.) \(E D \subset P I D \subset U F D\)
(d.) \(P I D \subset U F D \subset E D\)

16. Let \(I_{1}\) be the ideal generated by \(x^{4}+3 x^{2}+2\) and \(I_{2}\) be the ideal generated by \(x^{3}+1\) in \(\mathbb{Q}[x]\). If \(F_{1}=\frac{\mathbb{Q}[x]}{I_{1}}\) and \(F_{2}=\frac{\mathbb{Q}[x]}{I_{2}}\), then

(a.) \(F_{1}\) and \(F_{2}\) are fields
(b.) \(F_{1}\) is a field, but \(F_{2}\) is not a field
(c.) \(F_{1}\) is not a field while \(F_{2}\) is a field
(d.) Neither \(F_{1}\) nor \(F_{2}\) is a field.

17. For which of the following primes \(p\), does not the polynomial \(x^{4}+x+6\) have a root of multiplicity \(>1\) over a field of characteristic \(p\) ?

(a.) \(p=2\)
(b.) \(p=3\)
(c.) \(p=5\)
(d.) \(p=7\)

18. Which of the following statements is true about \(S=\mathbb{Z}[x]\) ?

(a.) \(\mathrm{S}\) is an Euclidean domain since all its ideals are principal
(b.) \(\mathrm{S}\) is an Euclidean domain since \(\mathbb{Z}\) is an Euclidean domain
(c.) \(\mathrm{S}\) is not an Euclidean domain since \(\mathrm{S}\) is not even an integral domain
(d.) \(\mathrm{S}\) is not an Euclidean domain since it has non-principal ideals

19. For positive integers \(n\) and \(m\), where \(n, m>1\), suppose that \(n \mathbb{Z}\) and \(m \mathbb{Z}\) are isomorphic as rings. Then,

(a.) there is no restriction on \(n\) and \(m\)
(b.) \(n=m\)
(c.) g.c.d \((n, m)=1\)
(d.) necessarily \(n \mid m\) or \(m \mid n\), but not both

20. The number of roots of the polynomial \(x^{3}-x\) in \(\frac{\mathbb{Z}}{6 \mathbb{Z}}\) is

(a.) 1
(b.) 2
(c.) 3
(d.) 6