Practice Questions for CSIR NET Ring Theory : Field Theory III

21. Let \(\mathrm{K}\) be an extension of the field \(\mathrm{Q}\) of rational numbers

(a.) If \(K\) is a finite extension then it is an algebraic extension
(b.) If \(K\) is an algebraic extension then it must be a finite extension
(c.) If \(K\) is an algebraic extension then it must be an infinite extension
(d.) If \(K\) is a finite extension then it need not be an algebraic extension

22. What is the degree of the minimal splitting field of the polynomial \(f(x)=x^p-2\) over \(Q\) where \(\mathbb{R}\) is an odd prime?

(a.) \(p\)
(b.) \(p^2\)
(c.) \(p^2-p\)
(d.) \(p-1\)

23. Let \(G\) denote the group of all the automorphisms of the field \(F_{3^{100}}\) that consists of \(3^{100}\) elements. Then the number of distinct subgroups of \(G\) is equal to

(a.) 4
(b.) 3
(c.) 100
(d.) 9

24. Consider the algebraic extension \(E=Q(\sqrt{2}, \sqrt{3}, \sqrt{5})\) of the field \(\mathbb{Q}\) of rational numbers. Then \([E: Q]\), the degree of \(E\) over \(\mathrm{Q}\), is

(a.) 3
(b.) 4
(c.) 7
(d.) 8

25. Find the degree of the field extension \(\mathbf{Q}(\sqrt{2}, \sqrt[4]{2}, \sqrt[8]{2})\) over \(\mathbb{Q}\)

(a.) 4
(b.) 8
(c.) 14
(d.) 32

26. For a positive integer \(n\), let \(f_n(x)=x^{n-1}+x^{n-2}+\ldots+x+1\). Then

(a.) \(f_n(x)\) is an irreducible polynomial in \(\mathrm{Q}[x]\) for every positive integer \(n\)
(b.) \(f_p(x)\) is an irreducible polynomial in \(Q[x]\) for every prime number \(p\)
(c.) \(f_{p^e}(x)\) is an irreducible polynomial in \(\mathrm{Q}[x]\) for every prime number \(p\) and every positive integer \(e\)
(d.) \(f_p\left(x^{p^{e-1}}\right)\) is an irreducible polynomial in \(\mathbb{Q}[x]\) for every prime number \(p\) and every positive integer \(e\)

27. For a positive integer \(m\), let \(a_m\) denote the number of distinct prime ideals of the ring \(\frac{\mathrm{Q}[x]}{\left\langle x^m-1\right\rangle}\), Then

(a.) \(a_4=2\)
(b.) \(a_4=3\)
(c.) \(a_5=2\)
(d.) \(a_5=3\)

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

(a.) \(x^4+1\)
(b.) \(x^8+x^7+x^5-x^4+x^3+x-1\)
(c.) \(x^8-x^7+x^5-x^4+x^3-x+1\)
(d.) \(x^8-1\)

29. Which of the following integral domains are Euclidean domains?

(a.) \(\mathbb{Z}[\sqrt{-3}]=\{a+b \sqrt{-3}: a, b \in \mathbb{Z}\}\)
(b.) \(\mathrm{Z}[x]\)
(c.) \(\mathbb{R}\left[x^2, x^3\right]=\left\{f=\sum_{i=0}^n a_i x^i \in \mathbb{R}[x]: a_1=0\right\}\)
(d.) \(\left(\frac{Z[x]}{\langle(2, x)\rangle}\right)[y]\) Where \(x, y\) are independent variables and \(\langle(2, x)\rangle\) is the ideal generated by 2 and \(x\)

30. Let \(p\) and \(q\) be two distinct primes. Pick the correct statements from the following:

(a.) \(\mathbb{Q}(\sqrt{p})\) Is isomorphic to \(\mathbb{Q}(\sqrt{q})\) as fields.
(b.) \(\mathbb{Q}(\sqrt{p})\) is isomorphic to \(\mathbb{Q}(\sqrt{q})\) as vector spaces over \(Q\)
(c.) \([(\mathrm{Q}(\sqrt{p}),(\sqrt{q}): \mathrm{Q}]=4\)
(d.) \(\mathrm{Q}(\sqrt{p}, \sqrt{q})=\mathrm{Q}(\sqrt{p}+\sqrt{q})\)

31. Let \(\omega=\cos \frac{2 \pi}{10}+i \sin \frac{2 \pi}{10}\). Let \(K=\mathbb{Q}\left(\omega^2\right)\) and let \(L=\mathbb{Q}(\omega)\). Then

(a.) \([L: Q]=10
(b.) \([L: K]=2
(c.) \([K: Q]=4
(d.) \(L=K

32. Let \(K\) and \(F\) be field extension both of degree 3 over \(Q\). Let \(L=K . F\) be the composite, i.e. smallest field extension of 0 containing both \(K \& F\); then \([L: 0]\) could be

(a.) 3 or 9 ; but neither 5 nor 6
(b.) 9, but none of \(3,5,6\)
(c.) 3 or 6 or 9 , but not 5
(d.) Any of 3,5,6,9

33. Let \(F \subseteq E\) be a field extension. Let \(a \in E\) be a root of an irreducible polynomial \(f(x)\) over \(F\) multiplicity three. Let \(\beta\) be any other root of \(f(x)\) in \(E\). Then the multiplicity of \(\beta\) is

(a.) Two
(b.) Greater than three
(c.) Three
(d.) One

34. Let \(K=\mathrm{Q}(\sqrt{2}, i)\) be the field generated over Q by \(\sqrt{2}\) and \(i\). Then the dimension of \(\mathrm{Q}(\sqrt{2}, i)\), as a \(\mathrm{Q}\)-vector space is equal to

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