Algebra Test - T 2

Rsquared Mathematics

Name: *

Email: *

Phone: *

Please fill in all the required fields correctly.

Time: 45:00

Topic: Algebra

Note : Welcome to Rsquared. We will continue with such quizzes in the future and this is just beginning. We request you to forward this quiz to your knowns so that it can spread more.
The Quiz contains a total of 10 Questions.
Part B(Single Correct) contains 5 questions, each of 3 marks and -0.75 marks for incorrect answers.
Part C(MSQ) contains 5 questions, each with 4.75 marks and no negative marking. The maximum score for the Test is 38.75. Best of Luck.

Part - B.

1. Which of the following are correct -

(a.) \(\mathbf{Q}(\sqrt{2})\) and \(\mathbf{Q}(\sqrt{3})\) are isomorphic as fields
(b.) \(\mathbf{Q}(\sqrt{2})\) and \(\mathbf{Q}(i \sqrt{2})\) are isomorphic as fields
(c.) \(\mathbf{Q}\left(2^{1 / 4}\right)\) and \(\mathbf{Q}\left(2^{1 / 4}, 2^{1 / 2}\right)\) are isomorphic as rings.
(d.) None of these

Explanation: (c): \(\mathbf{Q}\left(2^{1 / 4}\right)=\mathbf{Q}\left(2^{1 / 4}, 2^{1 / 2}\right)\)
\( \because \mathbf{Q}\left(2^{1 / 4}\right) \subseteq \mathbf{Q}\left(2^{1 / 4}, 2^{1 / 2}\right)\)
Also \( 2^{1 / 4} \in \mathbf{Q}\left(2^{1 / 4}\right) \Rightarrow 2^{1 / 2} \in \mathbf{Q}\left(2^{1 / 4}\right)\)
\(\Rightarrow 2^{1 / 4}, 2^{1 / 2} \in \mathbf{Q}\left(2^{1 / 4}\right) \Rightarrow \mathbf{Q}\left(2^{1 / 4}, 2^{1 / 2}\right) \subseteq \mathbf{Q}\left(2^{1 / 4}\right)\)
\(\Rightarrow \mathbf{Q}\left(2^{1 / 4}\right)=\mathbf{Q}\left(2^{1 / 4}, 2^{1 / 2}\right)\)

2. The number of one-one homomorphisms from \(D_3 \times \mathbf{Z}_5\) to \(S_3 \times \mathbf{Z}_5\) is .

(a.) 1
(b.) 6
(c.) 8
(d.) 24

Explanation: \(D_3 \cong S_3\), so both groups are the same up to isomorphism. We are just asked about the number of automorphism on \(S_3 \times \mathbf{Z}_5\) also if \(G_1\) and \(G_2\) are two groups of coprime order then \(Aut(G_1 \times G_2)\cong Aut(G_1) \times Aut(G_2)\)

3. The number of normal subgroups in a non-abelian group of order 203 is

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

Explanation: 29 SSG is unique.

4. Which of the following is incorrect-

(a.) The set of all bijections on any non-empty set forms a group with respect to the composition of functions.
(b.) The set of all bijections on a non-empty finite set forms a group with respect to the composition of functions.
(c.) Every ring can be embedded into a ring with unity.
(d.) None of these.

Explanation: All options are correct.

5. Let \(G=\{a_1, a_2, \ldots, a_{20}\}\) be a group of order 20. For \(b, c \in G\), let \(b G=\{b a_1, b a_2, \ldots, b a_{20}\}\) and \(G c=\{a_1 c, a_2 c, \ldots, a_{20} c\}\). Then

(a.) \(b G=G c\) only if \(b=c\)
(b.) \(b G=G c\) for all \(b, c \in G\)
(c.) \(b G=G c\) only if \(G\) is abelian
(d.) \(b G \neq G c\) if \(b \neq c\)

Explanation: \(bG=G=Gc\)

Part - C

6. Let \(G\) be a group of order 99. Then

(a.) \(G\) has a subgroup of order 33.
(b.) \(G\) has a subgroup of order 3.
(c.) \(G\) has at least six normal subgroups.
(d.) \(G\) is always abelian.

Explanation: \(G\) is abelian.

7. Let \(G\) denote a group and \(A\) denote a non-empty set. A group action of \(G\) on \(A\) is a homomorphism from \(G\) to the group of bijections on \(A\). Let \(\eta\) denote the number of group actions from \(G\) to \(A\). Then which of the following combinations is/are true?

(a.) \(G=S_3\), \(A=K_4\), \(\eta=34\)
(b.) \(G=S_3\), \(A=\mathbf{Z}_2\), \(\eta=2\)
(c.) \(G=K_4\), \(A=\mathbf{Z}_3\), \(\eta=10\)
(d.) \(G=\mathbf{Z}_2\), \(A=\mathbf{Z}_3\), \(\eta=4\)

Explanation: For (a), we need to find the number of homomorphisms from \(S_3 \rightarrow \operatorname{sym}(K_4) \cong S_4\), which is 34, and the same process for all other options.

8. Which of the following is/are irreducible over \(\mathbb{Q}\)?

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

Explanation: (c): if possible reducible then \(\left(x^4-x^2+1\right)=\left(x^2+ax+1\right)\left(x^2+bx+1\right)\)
\(\Rightarrow a+b=0\) and \(ab=-2\)
which is impossible so it is irreducible

9. Let \(R=\mathbb{Q}[x]\) be the ring of the polynomial. Which of the following is/are true?

(a.) \(R\) has a finite number of nilpotent elements.
(b.) \(R\) has an infinite number of nilpotent elements.
(c.) \(R\) has infinitely many prime ideals.
(d.) \(R\) has a prime ideal that is not maximal.

Explanation: \(\{0\}\) is the prime ideal of \(\mathbb{Q}[x]\) but not maximal as \(\{0\} \subset \langle x \rangle \subset \mathbb{Q}[x]\) Exp.

10. Which of the following is correct?

(a.) Let \(G=D_n\) (Dihedral group) then every subgroup of \(G\) of odd order is cyclic.
(b.) \(\exists\) a non-trivial homomorphism from \(S_n (n \geq 3)\) to \(\left(\mathbb{R}^*\right)\)
(c.) If \(N(H)=\{x \in G: xH=Hx\}=G\) then \(H\) is normal in \(G\)
(d.) Every finite subgroup of \(\frac{\mathbb{Q}}{\mathbb{Z}}\) is cyclic

Explanation: a,b,c,d.