Practice Questions for CSIR NET Ring Theory : Field Theory

1. Let \(p(x)=9 x^5+10 x^3+5 x+15\) and \(q(x)=x^3-x^2-x-2\) be two polynomials in \(\mathrm{Q}[x]\). Then, over \(\mathbb{Q}\),

(a.) \(p(x)\) and \(q(x)\) are both irreducible
(b.) \(p(x)\) is reducible but \(q(x)\) is irreducible
(c.) \(p(x)\) is irreducible but \(q(x)\) is reducible
(d.) \(p(x)\) and \(q(x)\) are both reducible

2. Which of the following statements is false?

(a.) The polynomial \(x^2+x+1\) is irreducible in \(\mathbb{Z} / 2 \mathbb{Z}[x]\)
(b.) The polynomial \(x^2-2\) is irreducible in \(\mathrm{Q}[x]\)
(c.) The polynomial \(x^2+1\) is reducible in \(\mathbb{Z} / 5 \mathbb{Z}[x]\)
(d.) The polynomial \(x^2+1\) is reducible in \(\mathbf{Z} / \operatorname{TZ}\{x]\)

3. Let \(\mathrm{Q}\) be the field of rational number and consider \(\mathbf{Z}_2\) as a field modulo 2. Let \(f(x)=x^3-9 x^2+9 x+3\) Then \(f(x)\) is

(a.) irreducible over but reducible over \(\mathbb{Z}_2\)
(b.) irreducible over both \(\mathrm{Q}\) and \(\mathrm{Z}_2\)
(c.) reducible over \(\mathbb{Q}\) but irreducible over \(\mathbf{z}_2\)
(d.) reducible over both \(Q\) and \(Z_2\)

4. The polynomial \(x^3+5 x^2+5\) is

(a.) Irreducible over \(\mathbf{Z}\) but reducible over \(\mathrm{Z}_5\)
(b.) Irreducible over both \(\mathbf{Z}\) and \(\mathbb{Z}_5\)
(c.) Reducible over \(\mathbb{Z}\) but irreducible over \(\mathbf{Z}_5\)
(d.) Reducible over both \(\mathrm{Z}_{\text{and}} \mathrm{Z}_5\)

5. The polynomial \(f(x)=x^5+5\) is

(a.) Irreducible over \(\mathrm{C}\)
(b.) Irreducible over \(\mathbb{R}\)
(c.) Irreducible over \(\mathbb{Q}\)
(d.) Not irreducible \(\mathbb{Q}\)

6. The polynomial \(x^3-7 x^2+15 x-9\) is

(a.) Irreducible over both \(\mathbf{Z}\) and \(\mathbf{Z}_3\)
(b.) Irreducible over \(\mathbb{Z}\) but reducible over \(\mathbb{Z}_3\)
(c.) Reducible over \(\mathbb{Z}\) but irreducible over \(\mathbf{Z}_3\)
(d.) Reducible over both \(\mathbb{Z}\) and \(\mathbb{Z}_3\)

7. Let \(f(x)=x^3+2 x^2+1\) and \(g(x)=2 x^2+x+2\) . Then over \(\mathbb{Z}_3\),

(a.) \(f(x)\) and \(g(x)\) are irreducible
(b.) \(f(x)\) is irreducible, but \(g(x)\) is not
(c.) \(g(x)\) is irreducible, but \(f(x)\) is not
(d.) Neither \(f(x)\) nor \(g(x)\) is irreducible.

8. Let \(\langle p(x)\rangle\) denote the ideal generated by the polynomial \(p(x)\) in \(\mathbb{Q}[x]\). If \(f(x)=x^3+x^2+x+1\) and \(g(x)=x^3-x^2+x-1\), then

(a.) \(\langle f(x)\rangle+\langle g(x))=\left\langle x^3+x\right\rangle\)
(b.) \(\langle f(x)\rangle+\langle g(x)\rangle=\langle f(x) \cdot g(x)\rangle\)
(c.) \(\langle f(x)\rangle+\langle g(x)\rangle=\left\langle x^2+1\right\rangle\)
(d.) \(\langle f(x)\rangle+\langle g(x)\rangle=\left\langle x^2-1\right\rangle\)

9. Determine which of the following polynomials are irreducible over the indicated rings.

(a.) \(x^5-3 x^4+2 x^3-5 x+8\) over \(\mathbb{R}\)
(b.) \(x^3+2 x^2+x+1\) over \(\mathbf{Q}\)
(c.) \(x^3+3 x^2-6 x+3\) over \(\mathbb{Z}\)
(d.) \(x^4+x^2+1\) over \(\mathbb{Z} / 2 Z\)

10. In which of the following fields, the polynomial \(x^3-312312 x+123123\) is irreducible in \(\mathbb{F}[x]\)?

(a.) The field \(\mathbb{F}_3\) with 3 elements
(b.) The field \(\mathbb{F}_7\) with 7 elements
(c.) The field \(\mathbb{F}_{13}\) with 13 elements
(d.) The field \(\mathbb{Q}\) of rational numbers