Solve your IGNOU Doubts

Question:

Suppose that $M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$ 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:

A^tD - C^tB 1

A^tC - C^tA = 0

B^tD-D^tB = 0

(Hint: Use block matrix multiplication.)

Also, check that the matrix $\begin{bmatrix}0&-A\\A&0\end{bmatrix}$ where A is a n x n orthogonal matrix, is a symplectic matrix

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Question:

If Fis a finite field show that there is always an irreducible polynomial of the form x³ - x + a where a ∈ 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 x⁹ - x ∈ $\mathbb{F}_{3}$ [x] guess the number of irreducible polynomials of degree 2 over $\mathbb{F}_{3}$ . Find all the irreducible polynomials of degree 2 over $\mathbb{F}_{3}$ .

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Question:

If char(F) ≠ 2, show that a polynomial ax² + bx + c is irreducible iff $b^2-4ac\notin\mathbb{F}^{*2}$ where $\mathbb{F}^{*2}$ is the group of squares in $\mathbb{F}$ .

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Question:

Find $[\mathbb{Q}(\sqrt[3]{2},\omega):\mathbb{Q}]$ where ?³ = 1, ? ≠ 1.

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Question:

Let ? ⊂ ? and let ?, ? ∈ ? be algebraic over F of degree m and n, respectively. Show that [? (?, ?) ∶ ? ] ≤ ??. What can you say about [? (?, ?) ∶ ? ] if m and n are coprime?

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Question:

Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?²) = ? (?).

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Question:

Compute the following:

(i) $\left(\frac{173}{211}\right)$ (ii) $\left(\frac{167}{239}\right)$

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Question:

Is there a finite group with class equation 1+1+2+2+2+2+2+2?

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Question:

Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.

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Question:

Describe the set of primes p for which x² - 11 splits into linear factors over Zp.

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Question:

Find the elementary divisors and invariant factors of $\mathbb{Z}_8\times\mathbb{Z}_{12}\times\mathbb{Z}_{15}$

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Question:

Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that Pis normal in K.

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Question:

Consider the natural action of GL₂ $\left(\mathbb{R}\right)$ on M₂ $\left(\mathbb{R}\right)$ , the set of 2 x 2 real matrices, by left multiplication.

(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M₂ $\left(\mathbb{R}\right)$ is {I}, where I is the 2 x 2 identity matrix.

(ii) Suppose that det(x) = 0 in the remaining parts of this exercise. We will show that the stabiliser of x is infinite. If x = 0, the stabiliser of x is GL₂ $\left(\mathbb{R}\right)$ . So suppose x ≠ 0. Let us write $X=\begin{bmatrix}a&c\\b&d\end{bmatrix}.$ Then, $\begin{bmatrix}a\\b\end{bmatrix}=\lambda\begin{bmatrix}c\\d\end{bmatrix}$ for non-zero λ ∈ R. Why?

(iii) Let $\begin{bmatrix}a'\\b'\end{bmatrix}$ be a vector that is not a scalar multiple of $\begin{bmatrix}a\\b\end{bmatrix}$ . Show that there is a matrix b = $\begin{bmatrix}u&v\\w&z\end{bmatrix}$ such that b $\begin{bmatrix}a\\b\end{bmatrix}$ = 0 and b $\begin{bmatrix}a'\\b'\end{bmatrix}$ = α $\begin{bmatrix}a'\\b'\end{bmatrix}$ (Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)

(iv) Check that I-b is in the stabiliser of x. Also, show that there are infinitely many choices of a for which I - b is invertible.

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Question:

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 G_s1 = G_s2 for all s₁, $₂ ∈ X.

(b) There are exactly 8 elements of order 3 in S₄.

(d) $F_7(\sqrt{3})=F_7(\sqrt{5})$

(e) For any $a\in\mathbb{F}_{2^5}^*,\quad a\ne 1,\quad\mathbb{F}_{2^5}=\mathbb{F}_2[a]$

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Question:

Evaluate the following integrals

a) $\int_0^\infty\frac{x^2+2}{(x^2+1)(x^2+4)}\,dx$

b) $\int_{-\infty}^{\infty}\frac{\sin^2 2x}{1+x^2}\,dx$

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Question:

Show that there is only one linear fractional transformation that maps three given distinct points, $z_{1},z_{2}$ and $z_{3}$ in the extended $z$ plane onto three specified distinct points $w_{1},w_{2}$ , and w,₃ in the extended w plane.

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Question:

Find the image of the semi-infinite strip $x>0,\quad 0<y<1$ when $w=l/z.$ . Sketch the strip and its image.

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Question:

22-1 Prove that the linear fractional transformation $\phi(z)=\frac{2z-1}{2-z}$ maps the circle $c:|z|=1$ into itself. Also prove that $f\left(z\right)$ is conformal in $\overline{D}=\{z:|z|\leq 1\}.$