Basic Algebra test

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  1. Every finite integral domain is a field.
    • True
    • False
  2. Which of the following is an irreducible polynomial over the integers
    • (x + 1)(x + 4)
    • x^3 + 10x^2 + 8x + 2
    • 2x^3 + 20x^2 + 16x + 4
    • 5
  3. Let f: G ---> H be a homomorphism between groups (H is an abelian group). Which of the following is not a true statement?
    • The kernel of f is a normal subgroup of G.
    • f(G) is an abelian subgroup of H.
    • G is an abelian group.
    • f^-1 (H) is a normal subgroup of G.
  4. How many integer solutions does 5x+10y+20z = 12 have?
    • 3.
    • 1.
    • 0.
    • Infinite.
  5. Let F be a field. Let V be a F-vectorial space of dimension 3. Let W be a F-vectorial space of dimension 4. Let X be the set of linear transformations from V to W, X is a F-vectorial space. What is the dimension of X?
    • 3
    • 4
    • 7
    • 12
    • Infinite.
  6. How many fields lies between the reals and the complex numbers?
    • 0.
    • 1.
    • 4.
    • Infinite.
  7. Let G be a group with 6 elements. Which of the following can't be true.
    • There exists a subgroup of G with 1 element.
    • There exists a subgroup of G with 2 elements.
    • There exists a subgroup of G with 3 elements.
    • There exists a subgroup of G with 4 elements.
  8. Let C be the set of 2x2 matrices of the form (a -b;b a) (a,b real numbers). Which of the following is false:
    • If A,B are matrices in C, then AB = BA.
    • If I is the identity matrix, and 0 is the zero matrix, then there exists X in C such that X*X + I = 0.
    • There exists a non-zero matrix in C which is not invertible.
    • The dimension of C as a real vector space is 2.
  9. Let p be a prime number. Let G be a group of order p^2. Which of the following must be true.
    • G is abelian.
    • In G there is an element of order p^2.
    • There is a proper nontrivial normal subgroup of G.
    • None of the above are true.
  10. Every field has a subfield isomorphic to the rational numbers, or there exist a prime number p such that theres a subfield isomorphic to Z/pZ
    • True.
    • False.

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