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MA232 Final Exam - Abstract Algebra II
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MA232 Final Exam - Abstract Algebra II
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25 Questions

1. Fill in the blank. The rank of a linear map is also the dimension of the map's _______________.
2. If f(x) is an irreducible polynomial, with f(x) | p(x)q(x), then which of the following is true?
3. Which of the following statements about the Galois group of E over F is true?
4. Which of the following is true of a ring with a zero divisor?
5. Suppose E is a field extension of F and further suppose there existed an element a in E such that F(a) = E.What is such an element called?
6. Let X = {1, 2, 3, 4, 6, 12, 24} be the set of divisors of 24, and let M = {2,3,4,6} be a proper subset of X.What is the least upper bound of M?
7. Suppose G is some cyclic group order n. If we know for some a ∈ G that a5 = e and a11 = e, then what is the smallest order G may take?
8. What is the set of all automorphisms of some field extension E of a field F that fix F elementwise called?
9. Suppose G = {a,b,c,d,e,f,g,h,i,j} is a group.What are the only possible orders of subgroups of G?
10. G is a finite group with proper, nontrivial subgroups H and J. Which of the following statements must be true?
11. Let f be a polynomial with rational coefficients and let it be factorable into two polynomials p and q.If deg p(x) = j and deg q(x) = k, then what must the degree of f be?
12. Fill in the blank. Let R be a Boolean ring. Then, for every a in R, a2 = a. Then, R must be ________________.
13. BA field is a commutative division ring.
14. Suppose f(x) is a polynomial with rational coefficients.If f(x) can be expressed as the product of two non-constant polynomials p and q, then what must be true of p and q?
15. Suppose G is a group of order 98.Which of the following statements must be true?
16. Suppose some subgroup H of G has order 5 and suppose H ≠ G. If G is abelian, what is the smallest order G may have?
17. Suppose that we know some group G contains the elements {a, b, c, d, e}, where e is the identity.What is the largest subgroup of G?
18. Suppose G is a group of order 21, and H is a proper subgroup of order 3.How many distinct left cosets of H are there?
19. Let G be a finite group with proper subgroups H, J and K. Further, let K ⊂ J ⊂ H. If [G:H] = 5, [H:J] = 3 and [J:K] = 7, what is [G:K]?
20. Let G be a finite group of order 63. If H and J are proper, nontrivial subgroups of G, H ≠ J, what is the largest order H ∩ J may have?
21. Which of the following statements is true of a vector homomorphism?
22. Suppose L is a lattice.Then, which of the following is true?
23. Suppose G is an abelian group of order 270.Which of the following statements is false?
24. What is a vector homomorphism that is one-to-one and onto called?
25. If F is an integral domain and every polynomial with coefficients in F can be factored in distinct ways, then what is true of F?

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