Find where ?3 = 1, ? ≠ 1.
To solve the given problem, we need to compute the degree of the field extension \(\mathbb{Q}(\sqrt[3]{2}, \omega)\) over \(\mathbb{Q}\), where \(\omega\) is a cube root of unity (i.e., \(\omega^3 = 1\), and \(\omega \neq 1\)). ### Step 1: Understanding the field \(\mathbb{Q}(\sqrt[3]{2})\) The element \(\sqrt[3]{2}\) _____ _____ _______ ____ _____ _____.
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Find all the Sylow 2-subgroups of Dn, when n is odd. Describe them in terms of x and y.
symplectic matrix with as the first column . (Hint: For any matrix A, what is
?)
(iii) Complete the proof by showing that, given any non-zero vector non-zero vector, there is always a
such that
is symplectic.
Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then Gs1 = Gs2 for all s1, $2 ∈ X.
(b) There are exactly 8 elements of order 3 in S4.
(c)
(d)
(e) For any
Suppose that is 2n × 2n matrix where A, B, C and D are nxn matrices. Show that M is symplectic if and only if the following conditions are satisfied:
AtD - CtB 1
AtC - CtA = 0
BtD-DtB = 0
(Hint: Use block matrix multiplication.)
Also, check that the matrix where A is a n x n orthogonal matrix, is a symplectic matrix
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
Compute the following:
(i) (ii)
Show that is not a UFD by giving two different factorisations of 20.
| Question no. | Block 1 | Block 2 | Block 3 | Block 4 | Block 5 |
| 2 a) | 5 | ||||
| 2 b) | 3 | ||||
| 2 c) | 2 | ||||
| 3 c) | 10 | ||||
| 4 a) | 4 | ||||
| 4 b) | 3 | ||||
| 4 c) | 3 | ||||
| 5 a) | 2 | ||||
| 5 b) | 5 | ||||
| 5 c) | 3 | ||||
| 6 a) | 2 | ||||
| 6 b) | 6 | ||||
| 6 c) | 2 | ||||
| 7 a) | 4 | ||||
| 7 c) | 1 | ||||
| 7 c) | 3 | ||||
| 7 c) | 2 | ||||
| 8 a) | 3 | ||||
| 8 b) | 4 | ||||
| 8 c) | 2 | ||||
| 8 d) | 3 | ||||
| 8 e) | 3 | ||||
| 9 a) | 5 | ||||
| 9 b) | 5 | ||||
| 9 c) | 5 | ||||
| Total | 30 | 17 | 23 | 20 | 0 |