Solve your IGNOU Doubts

Question:

Let $X = {f \in C[0,1]: f(0) = 0}$

$Y = \begin{Bmatrix} g \in x: \int_{0}^{1}g(t)dt = 0 \end{Bmatrix}$

Prove that Y is a proper subspace of X. Is Y a closed subspace of X? Justify your answer

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Question:

Show that $Q(\sqrt{-19})$ is not a UFD by giving two different factorisations of 20

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Question:

Solve the following set of congruences:

$x \equiv 2 (mod \: 17)$

$3x \equiv 4 (mod \: 19)$

$x \equiv 7 (mod \: 23)$

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Question:

Let $G = \langle a,b\mid a^2,b^3,aba^{-1}b^{-1} \rangle$ . Show that G is the cyclic group of order six.

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Question:

In this exercise, we ask you to find the Sylow p-subgroups of the dihedral group $D_n = {x,y :x^n,y^2,yxyx}, n \in N, n\geq 2$

(a) Let p be an odd prime that divides n, $n = p^rl, p \nmid l$ . Suppose $C = \langle x^l \rangle$ . Show that C is the unique Sylow p-subgroup of D_n .

(b) Prove the relation $y^ix^jy^kx^l = \left\{\begin{matrix} y^ix^{j+1} & if\: k\: is \: \: even \\ y^{i+k}x^{l-j} & if\: k\: is \: \: odd \end{matrix}\right.$

Further, find all the elements of order 2 in D_n .

(d) Suppose n is even, n = 2^km, where $2 \nmid m, k\geq 2$ . Let $N = \langle x^m \rangle$ and $H = \langle y \rangle$ . Show that HN is a subgroup of D_n . What is its order?

(e) Suppose n is as in the previous part. Find all the Sylow 2-supgroups of D_n . Describe them in terms of x and y.

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Question:

Complete the proof by showing that, given any non-zero vector $\begin{bmatrix} a \\ b \end{bmatrix}$ , there is always a non-zero vector $\begin{bmatrix} a' \\ b' \end{bmatrix}$ such that $\begin{bmatrix} a&a' \\ b&b' \end{bmatrix}$ is symplectic.

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Question:

Show that, to prove that SP₂(R) acts transitively on GL₂(R), it is enough to show that, for any vector $\begin{bmatrix} a\\ b \end{bmatrix} \neq 0 \in R^2$ , there is a $2\times 2$ symplectic matrix with $\begin{bmatrix} a \\ b \end{bmatrix}$ as the first column. (Hint: For any matrix A, what is $A\begin{bmatrix} I \\ 0 \end{bmatrix}$ ?)

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Question:

Show that a matrix $\begin{bmatrix} a &b \\ c& d \end{bmatrix} \in GL_2(R)$ is symplectic if and only if ad - bc = 1

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Question:

The aim of this exercise is to show that SP₂(R)acts transitively on R² \ {0}.

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Question:

Suppose that $m = \begin{bmatrix} A & B\\ C & D \end{bmatrix}$ is $2n\times 2n$ matrix where A,B,C and D are $n\times n$ matrices. Show that M is symplectic if and only if the following conditions are satisfied:

$A^tD - C^tB = I$

$A^tC - C^tA = 0$

$B^tD - D^tB = 0$

(Hint: Use block matrix multiplication.) Also, check that the matrix $B^tD - D^tB = 0\begin{bmatrix} 0 & -A\\ A& 0 \end{bmatrix}$ , where A is a $n\times n$ orthogonal matrix, is a symplectic matrix.

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Question:

If F is a finite field show that there is always an irreducible polynomial of the form $x^3-x+a$ where $a \in F$ .(Hint: Show that $x \mapsto x^3 - x$ is not a surjective map.)

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Question:

By looking at the factorisation of $x^9 - x \in F_3[x]$ guess the number of irreducible polynomials of degree 2 over $F_3$ . Find all the irreducible polynomials of degree 2 over $F_3$ .

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Question:

If $char(F) \neq 2$ , show that a polynomial $ax^2 + bx + c$ is irreducible iff $b^2-4ac \notin F^{*2}$ where $F^{*2}$ is the group of squares in $F^{*}$ .

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Question:

Find $[Q(\sqrt[3]{2},\omega)\vdots Q]$ where $\omega^3 = 1, \omega \neq 1$

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Question:

Let $F \subset K$ and let $\alpha ,\beta \in K$ be algebraic over F of degree m and n, respectively. Show that $[F (\alpha ,\beta) : F]\leq mn$ . What can you say about $[F (\alpha ,\beta) : F]$ , if m and n are coprime?