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

31. The polynomial ring \(\mathbb{Z}[x]\) is

(a.) A Euclidean domain but not a PID
(b.) A PID but not Euclidean
(c.) Neither PID nor Euclidean
(d.) Both PID and Euclidean

32. The number of non-trivial ring homomorphisms from \(\mathbb{Z}_{12}\) to \(\mathbb{Z}_{28}\) is

(a.) 1
(b.) 3
(c.) 4
(d.) 7

33. Let \(I\) denote the ideal generated by \(x^{4}+x^{3}+x^{2}+x+1\) in \(\mathbb{Z}_{2}[x]\) and \(F=\mathbb{Z}_{3}[x] / I\). Then,

(a.) \(F\) is an infinite field
(b.) \(F\) is a finite field of 4 elements
(c.) \(F\) is a finite field of 8 elements
(d.) \(F\) is a finite field of 16 elements

34. The polynomial \(f(X):=X^{2}+a X+1\) in \(\mathbb{Z}_{3}[X]\) is

(a.) irreducible in \(\mathbb{Z}_{3}[X]\)
(b.) irreducible in \(\mathbb{Z}_{3}[X]\) if and only if \(a=0\)
(c.) irreducible in \(\mathbb{Z}_{3}[X]\) if and only if \(a=1\)
(d.) always reducible in \(\mathbb{Z}_{3}[X]\)

35. Let \(\mathbb{F}_{p}\) denote the field \(\frac{\mathbb{Z}}{p \mathbb{Z}}\), where \(p\) is a prime. Let \(\mathbb{F}_{p}[x]\) be the associated polynomial ring. Which of the following quotient rings are fields?

(a.) \(\frac{\mathbb{F}_{5}[x]}{\left\langle x^{2}+x+1\right\rangle}\)
(b.) \(\frac{\mathbb{F}_{2}[x]}{\left\langle x^{3}+x+1\right\rangle}\)
(c.) \(\frac{\mathbb{F}_{3}[x]}{\left\langle x^{3}+x+1\right\rangle}\)
(d.) None of these

36. Let \(\mathrm{R}\) be a commutative ring and \(R[x]\) be the polynomial ring in one variable over \(R\).

(a.) If \(\mathrm{R}\) is a U.F.D., then \(R[x]\) is a U.F.D.
(b.) If \(\mathrm{R}\) is a P.I.D, then \(R[x]\) is a P.I.D.
(c.) If \(\mathrm{R}\) is an Euclidean domain, then \(R[x]\) is an Euclidean domain
(d.) If \(\mathrm{R}\) is a field, then \(R[x]\) is an Euclidean domain.

37. Let \(F\) and \(F^{\prime}\) be two finite fields of order \(q\) and \(q\) respectively. Then:

(a.) \(F^{\prime}\) contains a subfield isomorphic to \(F\) if and only if \(q \leq q^{\prime}\).
(b.) \(F^{\prime}\) contains a subfield isomorphic to \(F\) if and only if \(q\) divides \(q^{\prime}\).
(c.) If the g.c.d of \(q\) and \(q^{\prime}\) is not 1 , then both are isomorphic to subfields of some finite field \(L\).
(d.) Both \(F\) and \(F^{\prime}\) are quotient rings of the ring \(\mathbb{Z}[X]\).

38. Consider the ring \(R=\mathbb{Z}[\sqrt{-5}]\) \(=\{a+b \sqrt{-5}: a, b \in \mathbb{Z}\}\) and the element \(\alpha=3+\sqrt{-5}\) of \(R\). Then

(a.) \(\alpha\) is prime
(b.) \(\alpha\) is irreducible
(c.) \(R\) is not a unique factorization domain
(d.) \(R\) is not an integral domain.

39. Let \(f(x)=x^{3}+x^{2}+x+1\) and \(g(x)=x^{5}+1\). Then in \(\mathbb{Q}[x]\)

(a.) g.c.d \((f(x), g(x))=x+1\)
(b.) g.c.d \((f(x), g(x))=x^{2}-1\)
(c.) l.c.m. \((f(x), g(x))=x^{5}+x^{3}+x^{2}+1\)
(d.) l.c.m. \((f(x), g(x))=x^{5}+x^{4}+x^{3}+x^{2}+1\)

40. Let \(\mathbb{R}[x]\) be the polynomial ring over \(\mathbb{R}\) in one variable. Let \(I \subseteq \mathbb{R}[x]\) be an ideal. Then

(a.) \(I\) is a maximal ideal if and only if \(I\) is a non-zero prime ideal
(b.) \(I\) is a maximal ideal if and only if the quotient ring \(\mathbb{R}[x] / I\) is isomorphic to \(\mathbb{R}\)
(c.) \(I\) is a maximal ideal if and only if \(I=\langle f(x)\rangle\), where \(f(x)\) is non-constant irreducible polynomial over \(\mathbb{R}\)
(d.) \(I\) is a maximal ideal if and only if there exists a non-constant polynomial \(f(x) \in I\) of degree 3