Tribhuvan University BSc Fourth Year Modern Algebra Exam Question 2082

Tribhuvan University · Modern Algebra (BSc Fourth Year) · 2082 BS · Annual Examination — 10 questions — a free old question paper on Goodone.club’s exam-paper archive.

  1. Define a binary operation on a set S. State whether the set ℤ of integers with usual multiplication forms a group, and justify your answer. Let ℚ⁺ be the set of positive rational numbers. For a, b ∈ ℚ⁺, define a ∗ b = ab/2. Does (ℚ⁺, ∗) form a group? Under this operation, solve 3 ∗ x = 6. [2+2+5+1] [10 marks]
  2. Define a permutation group and an isomorphism between groups. Prove that a group is isomorphic to a group of permutations. Write a group that is isomorphic to the group {1, ω, ω²}, where ω is a cube root of unity. How do they correspond? [1+1+6+1+1] [10 marks]
  3. Define a homomorphism between groups, and define the kernel and image of a homomorphism. Let ℂ be the set of complex numbers and let φ: ℂ → ℂ be defined by φ(x + iy) = y for every x + iy ∈ ℂ. Is φ a homomorphism? If so, find its kernel and image. Prove that a homomorphism between groups is injective if and only if its kernel contains the identity only. [2+2+2+4] OR Let G be a group, H a normal subgroup of G, and σ: G → G/H be defined by σ(a) = aH for a ∈ G. Prove that: (i) G/H is a group; (ii) σ is a homomorphism onto G/H whose kernel is H. [5+5] [10 marks]
  4. Define a ring and a field. Let ℚ be the set of rational numbers with the usual addition and multiplication. Show that ℚ(√2) = {a + b√2 : a, b ∈ ℚ} forms a field. [4+1+5] [10 marks]
  5. Let R and R′ be rings and let φ: R → R′ be a ring homomorphism with kernel I. Prove that: (i) φ carries zero to zero; (ii) φ(−a) = −φ(a) for every a ∈ R; (iii) I is an ideal of R. [3+3+4] [10 marks]
  6. Why do we extend a field? Justify with a concrete example. Let F be a field and let f(x) ∈ F[x] be a non-constant polynomial. Prove that there is an extension field E such that α ∈ E and f(α) = 0. [3+7] OR Explain why √2 and π are algebraic and transcendental, respectively. Let K be an extension field of a field F. Prove that α ∈ K is algebraic over F if and only if [F(α):F] is finite. [2+8] [10 marks]
  7. Let H and K be normal subgroups of a group G with K ≤ H. Prove that (G/K)/(H/K) ≅ G/H. Verify the result in the group of integers, taking the multiples of 6 and the multiples of 2 as subgroups. [5+5] OR Discuss the Sylow subgroups of a group of order 15 and, using Sylow's theory, prove that this group is cyclic. [4+6] [10 marks]
  8. Define a Gaussian integer and its norm. List the properties of the norm. Prove that the set ℤ[i] of Gaussian integers is an integral domain. Does it form a Euclidean domain? If so, explain how. [1+1+3+4+1] [10 marks]
  9. Write the relations between the roots and coefficients of f(x) = x⁴ + p₁x³ + p₂x² + p₃x + p₄ = 0. Are these relations sufficient to solve the equation? Justify your answer. Solve 2x³ + x² − 7x − 6 = 0, given that the difference between two of the roots is 3. [2+2+6] [10 marks]
  10. Define a reciprocal equation. Describe its form and explain how to recognize an equation as reciprocal. Solve 2x⁵ − 15x⁴ + 37x³ − 37x² + 15x − 2 = 0. OR Explain Cardano's method for solving a cubic equation. Solve 9x³ − 6x² + 1 = 0. [5+5] [10 marks]