1st Honours Algebra Quick Quiz 4 Click on the correct answer to the question. 1. The Root Theorem states that for f in F[x] and a in F. a is a root of f if and only if A. f divides (x - a) B. (x - a) divides f C. f(a) = 0 D. f(a) = a 2. A corollary to the Root Theorem states that for f in F[x] with deg(f) = n >= 0. Then f has A. at most n-1 distinct roots B. at least one root C. roots D. at most n distinct roots 3. Which of the following polynomials in Z2[x] are irreducible: A. x2 + 1 B. x3 + 1 C. x2 + x D. x2 + x + 1 4. In which of the following polynomial rings is x2 + 1 irreducible: A. C[x] B. Z2[x] C. Z3[x] D. R[x] 5. A group is a set with an operation such that the following four axioms hold, except that one of them is stated incorrectly, which one: A. If x, y are in G then xy is in G B. (xy)z = x(yz) , for all x, y, z in G C. there exists an element e such that xe = ex for all x in G D. for all x in G there is an element x' such that xx' = e 6. One of the following is not a group, which one: A. R under multiplication B. Z under addition C. Q under addition D. Zp under multiplication, where p is a prime 7. A group G is said be Abelian when A. G has two operations B. |G| is finite C. xy = yx , for all x, y in G D. xy = yx , for some x, y in G 8. The dihedral group of symmetries of a regular 3-gon (an equilateral triangle) has six elements. How many elements are there in the group of symmetries of a regular 4-gon (a square)? A. 8 B. 6 C. 7 D. 4! Label the vertices of an equilateral triangle with the numbers 1, 2, 3. The symmetries of the triangle can then be represented by the permutations of the set {1, 2, 3}. So for example the identity symmetry is (1)(2)(3), a rotation through 120 degrees by (123). 9. What then is the permutation that corresponds to a rotation through 240 degrees? A. (1)(23) B. (132) C. (231) D. (1)(2)(3) 10. What then is the permutation that corresponds to a reflection in the axis through the vertex 2 and the midpoint of the opposite side? A. (2)(13) B. (32)(1) C. (321) D. (1)(2)(3) 11. Two groups G1 and G2 are said to be isomorphic if there is a bijection f from G1 to G2, and A. f(xy) = f(y)f(x) B. f(xy) = f(x)f(y) C. f(x+y) = f(x)f(y) D. f(x) = f(y) implies x = y 12. Which of the following is a group: A. Zn, for any n >= 1, under multiplication B. Z under multiplication C. Sn under composition of permutations D. F[x], where F is a field 13. Which of the following is a true statement about group isomorphisms: A. Z3 (under addition) is isomorphic to Z4 (under addition) B. Z (under addition) is isomorphic to R (under addition) C. S3 (under permutation composition) is isomorphic to Z6 (under addition) D. Zp\{0} (under multiplication) is isomorphic to Up (under multiplication) 14. What is the order of the element 8 in Z12 (under addition) A. 1 B. 12 C. 3 D. 0 15. What is the order of the permutation (123)(4567) in S7 A. 3 B. 4 C. 6 D. 12 If you think I've got an answer wrong, e-mail me at rbreams@vcu.edu so that I can correct it.
Click on the correct answer to the question.
1. The Root Theorem states that for f in F[x] and a in F. a is a root of f if and only if A. f divides (x - a) B. (x - a) divides f C. f(a) = 0 D. f(a) = a
2. A corollary to the Root Theorem states that for f in F[x] with deg(f) = n >= 0. Then f has A. at most n-1 distinct roots B. at least one root C. roots D. at most n distinct roots
3. Which of the following polynomials in Z2[x] are irreducible: A. x2 + 1 B. x3 + 1 C. x2 + x D. x2 + x + 1
4. In which of the following polynomial rings is x2 + 1 irreducible: A. C[x] B. Z2[x] C. Z3[x] D. R[x]
5. A group is a set with an operation such that the following four axioms hold, except that one of them is stated incorrectly, which one: A. If x, y are in G then xy is in G B. (xy)z = x(yz) , for all x, y, z in G C. there exists an element e such that xe = ex for all x in G D. for all x in G there is an element x' such that xx' = e
6. One of the following is not a group, which one: A. R under multiplication B. Z under addition C. Q under addition D. Zp under multiplication, where p is a prime
7. A group G is said be Abelian when A. G has two operations B. |G| is finite C. xy = yx , for all x, y in G D. xy = yx , for some x, y in G
8. The dihedral group of symmetries of a regular 3-gon (an equilateral triangle) has six elements. How many elements are there in the group of symmetries of a regular 4-gon (a square)? A. 8 B. 6 C. 7 D. 4! Label the vertices of an equilateral triangle with the numbers 1, 2, 3. The symmetries of the triangle can then be represented by the permutations of the set {1, 2, 3}. So for example the identity symmetry is (1)(2)(3), a rotation through 120 degrees by (123).
9. What then is the permutation that corresponds to a rotation through 240 degrees? A. (1)(23) B. (132) C. (231) D. (1)(2)(3)
10. What then is the permutation that corresponds to a reflection in the axis through the vertex 2 and the midpoint of the opposite side? A. (2)(13) B. (32)(1) C. (321) D. (1)(2)(3)
11. Two groups G1 and G2 are said to be isomorphic if there is a bijection f from G1 to G2, and A. f(xy) = f(y)f(x) B. f(xy) = f(x)f(y) C. f(x+y) = f(x)f(y) D. f(x) = f(y) implies x = y
12. Which of the following is a group: A. Zn, for any n >= 1, under multiplication B. Z under multiplication C. Sn under composition of permutations D. F[x], where F is a field
13. Which of the following is a true statement about group isomorphisms: A. Z3 (under addition) is isomorphic to Z4 (under addition) B. Z (under addition) is isomorphic to R (under addition) C. S3 (under permutation composition) is isomorphic to Z6 (under addition) D. Zp\{0} (under multiplication) is isomorphic to Up (under multiplication)
14. What is the order of the element 8 in Z12 (under addition) A. 1 B. 12 C. 3 D. 0
15. What is the order of the permutation (123)(4567) in S7 A. 3 B. 4 C. 6 D. 12