Practice Questions for CSIR NET Ring Theory : Field Theory II

11. Consider the polynomial \(f(x)=x^4-x^3+14 x^2+5 x+16\). Also for a prime number \(p\), let \(\mathbb{F}_p\) denote the field with \(p\) elements. Which of the following is always true?

(a.) Considering \(f\) as a polynomial with coefficients in \(F_3\), it has no roots in \(F_3\).
(b.) Considering \(f\) as a polynomial with coefficients in \(F_3\), it is a product of two irreducible factors of degree 2 over \(\mathrm{F}_3\).
(c.) Considering \(f\) as a polynomial with coefficients in \(\mathbf{F}_7\), it has an irreducible factor of degree 3 over \(F_7\).
(d.) \(f\) is a product of two polynomials of degree 2 over \(\mathbb{Z}\).

12. Which of the following polynomials are irreducible in the ring \(\mathbb{Z}[x]\) of polynomials in one variable with integer coefficients?

(a.) \(x^2-5\)
(b.) \(1+(x+1)+(x+1)^2+(x+1)^3+(x+1)^4\)
(c.) \(1+x+x^2+x^3+x^4\)
(d.) \(1+x+x^2+x^3\)

13. Which of the polynomials are irreducible over the given rings?

(a.) \(X^5+3 X^4+9 X+15\) over \(\mathbb{Q}\), the field of rationals
(b.) \(X^3+2 X^2+X+1\) over \(\mathbb{Z} / 7 \mathbb{Z}\), the ring of integers modulo 7
(c.) \(X^3+X^2+X+1\) over \(\mathbb{Z}\), the ring of integers
(d.) \(X^4+X^3+X^2+X+1\) over \(\mathrm{Z}\), the ring of integers

14. Let \(f(x)=x^4+3 x^3-9 x^2+7 x+27\) and let \(p\) be a prime. Let \(f_p(x)\) denote the corresponding polynomial with coefficients in \(\mathbb{Z} / p \mathbb{Z}\). Then

(a.) \(f_2(x)\) is irreducible over \(Z / 2 Z\)
(b.) \(f(x)\) is irreducible over \(Q\)
(c.) \(f_3(x)\) is irreducible over \(\mathbb{Z} / 3 \mathbb{Z}\)
(d.) \(f(x)\) is irreducible over \(\mathrm{Z}\)

15. Which of the following is an irreducible factor of \(x^{12}-1\) over \(Q\)?

(a.) \(x^8+x^4+1\)
(b.) \(x^4+1\)
(c.) \(x^4-x^2+1\)
(d.) \(x^5-x^4+x^3-x^2+x-1\)

16. For the rings \(L=\frac{\mathbb{R}[x]}{\left\langle x^2-x+1\right\rangle}\); \(M=\frac{\mathbb{R}[x]}{\left\langle x^2+x+1\right\rangle}\); \(N=\frac{\mathbb{R}[x]}{\left\langle x^2+2 x+1\right\rangle}\)

Which one of the following is TRUE?

(a.) \(L\) is isomorphic to \(M\); \(L\) is not isomorphic to \(N\); \(M\) is not isomorphic to \(N\)
(b.) \(M\) is isomorphic to \(N\); \(M\) is not isomorphic to \(L\); \(N\) is not isomorphic to \(L\)
(c.) \(L\) is isomorphic to \(M\); \(M\) is isomorphic to \(N\)
(d.) \(L\) is not isomorphic to \(M\); \(L\) is not isomorphic to \(N\); \(M\) is not isomorphic to \(N\)

17. The degree of the extension \(Q(\sqrt{2}+\sqrt{3})\) over the field \(\mathbb{Q}(\sqrt{2})\) is

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

18. Let \(\mathrm{Q}\) denote the field of rational numbers, the ring \(Q[x] /\left\langle x^2+1\right\rangle\) is isomorphic to

(a.) The field of complex numbers.
(b.) \(\mathbb{Q}[x] /\left\langle x^2\right\rangle\)
(c.) \(\mathrm{Q}[x] /\left\langle\left(x^2+2 x+2\right)\right\rangle\)
(d.) None of the above.

19. Which of the following field extension is not a Galois extension?

(a.) \(\mathrm{Q}(\sqrt{2+\sqrt{2}}) / \mathrm{Q}\)
(b.) \(Q\left(2^{1 / 4}\right) / Q\)
(c.) \(I F_{p^*} \mid I F_p\)
(d.) \(\mathrm{Q}(\sqrt{2}, \sqrt{3}) \mid \mathrm{Q}\)

20. Let \(K\) be a field, \(L\) a finite extension of \(K\) and \(\mathrm{M}\) a finite extension of \(\mathrm{L}\). Then

(a.) \([M: K]=[M: L]+[L: K]\)
(b.) \([M: K]=[M: L][L: K]\)
(c.) \([M: L]\) divides \([M: K]\)
(d.) \([L: K]\) divides \([M: K]\)