Let “ ·n” be multiplication modulo n, and consider the set Un =
{ [a] ∈ Zn | there is a [b] ∈ Zn with [a] ·n [b] = [1]}
(a) Show that (Un, ·n ) is a group.
(b) Write down the Cayley table for U5. Hint: |U5| = 4.
(c) Write down the Cayley table for U12. Hint: |U12| = 4.
3. Suppose that a divide and conquer algorithm for
multiplication of n x n matrices is found such that it requires 6
multiplications and 31 additions of n/2 x n/2 submatrices. Write
the recurrence for the running time T(n) of this algorithm and find
the order of T(n).
Given n ∈N and p prime number and consider the polynomial f (x) = xn (xn-2)+1-p
1)Prove that f (x) is irreducible in Q [x]
2) If n = 1 and p = 3, find Q [x] / f (x))
3) Show that indeed Q [x] / (f (x)) is a field in the previous paragraph
PLEASE answer all subsections
Given two 2x2 matrices:
M=1
21
1,N=-1
2 1 -1
(a).Verify multiplication of M x N mod
26 = N x M mod 26 = I (identity matrix)
(b). Use Hill cipher to encrypt the
message EXAMS