1st Honours Algebra Quick Quiz 3

Click on the correct answer to the question.

1. Complete the statement of the following theorem: Let a, b, n be integers with n >= 1. The linear congruence
ax = b mod n has a solution x if and only if
A. gcd(a,b) | n
B. gcd(b,n) | a
C. gcd(a,n) | b

2.  From quick inspection, the solution x to the congruence 52x = 1 mod 53 is
A. 1
B. 52
C. 0

3. Complete the statement of the following theorem: Let r, n be integers with n >= 1. Then r is invertible in Z_n if
and only if gcd(r,n) =
A. r
B. n
C. 0
D. 1

4. In Z₇ the invertible elements are
A. 0,1,2,3,4,5,6
B. 1,2,3,4,5,6
C. 1
D. -1

5. U_n is the notation used for the elements in Z_n which are
A. nonzero
B. invertible
C. divide n

6. Fermat's Little Theorem is a corollary to
A. the Chinese Remainder Theorem
B. the Fundamental Theorem of Equivalence Relations
C. the Fundamental Theorem of Arithmetic
D. Euler's Theorem

7. Fermat's Little Theorem is the easiest way to see that
A. 3³⁰ = 1 mod 31
B. 1³⁰ = 1 mod 31
C. 2⁵ = 1 mod 31
D. 30 = -1 mod 31

8.  Let k be an integer k >= 1, and p a prime. Then the number of integers m such that 1 <= m <= p^k which
are relatively prime to p^k is
A. p^k
B. p^k - 1
C. p^k(1 - 1/p)
D. 1

9.  Which is the correct statement of the lemma which is used to prove the Chinese Remainder Theorem
A. Let a, b, c be integers with gcd(a,b) = 1. If a | c and b | c then ac | b
B. Let a, b, c be integers with gcd(a,b) = 1. If a | c and b | c then bc | a
C. Let a, b, c be integers with gcd(a,b) = 1. If a | c and b | c then a | c
D. Let a, b, c be integers with gcd(a,b) = 1. If a | c and b | c then ab | c

10. The Chinese Remainder Theorem was found in a book ~300 A.D. written by
A. Sun Zi
B. Zi Sun
C. Si Zun
D. Un Zit

11. An example of a field is
A. Z
B. the set of all 2x2 matrices with real entries
C. Z_n, for any positive integer n
D. Z_p, where p is a prime

12. An example of a commutative ring with 1 is
A. Z
B. the set of all 2x2 matrices with real entries
C. Sn, the set of all permutations of {1,2,3,..., n}

13. Let F be a field. The notation F[x] means
A. a function F depending on x
B. a field F with x in it
C. the set of all polynomials with coefficients from F
D. the set of all functions with coefficients from F

14. Let p and q be polynomials with coeffients from a field F then
A. deg(pq) <= max{deg(p), deg(q)}
B. deg(pq) = deg(p) + deg(q)
C. deg(pq) >= max{deg(p), deg(q)}
D. deg(p/q) = deg(p) - deg(q)

15. The degree of the zero polynomial is defined to be
A. zero
B. one
C. -infinity
D. undefined

16. A unit in a commutative ring with 1 is another word for
A. the number 1
B. a box you put a ring in
C. an invertible element

17. The Division Algorithm for polynomials states that: If a, b are polynomials with coefficients from a field F ,
and b is not the zero polynomial, then there exist
A. polynomials x, y with coefficients in F so that gcd(a,b) = ax + by
B. polynomials q, r with coefficients in F so that a = bq + r
C. unique polynomials q, r with coefficients in F so that a = bq + r
D. unique polynomials q, r with coefficients in F so that a = bq + r, where deg(r) < deg(b)

18. A polynomial with coefficients from a field F is said to be monic if
A. its leading coefficient is x
B. its leading coefficient is 1
C. it is nonzero
D. it is not a unit

If you think I've got an answer wrong, e-mail me at rbreams@vcu.edu so that I can correct it.