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MA231 Final Exam - Abstract Algebra I
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MA231 Final Exam - Abstract Algebra I
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25 Questions

1. Suppose G is an extension field over F with order 5 and F is an extension field over E with order 7. Then, G would be an extension field over E with what order?
2. Let f: A → B and g: B → C be maps. Which of the following statements is true?
3. Supposed K is a splitting field that has order 9 over some field F.What has to be true?
4. Let A = {a, b, c}, B = {d, e, f}, and C = {h, i, j} and suppose that f(a) = d, f(b) = f, f(c) = e, g(d) = i, g(e) = j, and g(f) = h. Which of the following represents (g°f)(x)?
5. Which of the following statements is true of a ring?
6. Which of the following statements about a ring with a zero divisor is true?
7. Let f: Z → Q be given by f(n) = n/1. What is true about f?
8. Suppose f: Q → Z given by f(p/q) = p. What is the most that can be said about f?
9. Which of the following statements is true?
10. Let E be the splitting field for a polynomial p(x) with coefficients in F of degree n. What must be true?
11. Which of the following is a group homomorphism?
12. The set F of all functions with the operator '+' is a group.Which of the following statements is true?
13. Let f be a ring isomorphism from ring R to ring T.Which of the following is true?
14. Fill in the blank. Let R be a Boolean ring. Then, for every a in R, a² = a. Then, R must be ______________.
15. What is the basis of the field extension Q(√3,√6) over Q?
16. Which of the following is a necessary condition for a function f to be invertible?
17. Which of the following is true about f: Z → Q?
18. Suppose G is a cyclic group with order n, and let |k| ≤ n. If we know ak = e, the identity element of G, then what must be true about k?
19. Let the center of G, Z(G) = {x ∊ G: xg = gx for all g ∊ G}. Which of the following statements is true?
20. If A = {a, b, c, d, e}, B = {a, c, e, f}, and C = {b, d, f, h}, which operation is equivalent to (A\ B) ∩ C?
21. Suppose G is a group, and we know for all g, h ∊ G, (gh)ⁿ = gⁿhⁿ . Then, which of following must be true?
22. If A = {a, b, c, d, e}, B = {a, c, e, f}, and C = {b, d, f, h}, what is the result of the operation (A\ B) ∩ C?
23. Which of the following statements is true of a subring of a commutative ring?
24. Which of the following represents a relation but not a function?
25. Suppose E is an extension field of order 2 over F, and K is an extension field of order 3 over E. Then, K is an extension field of what order over F?

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