HOMEWORK 1
3
(1) Given z = a + bi express the multiplicative inverse z −1 as c + di. That is,
find c and d in terms of a and b.
(2) Given z = a + bi and w = c + di, express the quotient z/w as e + f i. That
is find e and f in terms of a, b, c and d.
(3) Consider Z/3. This is the set {0, 1, 2} where we use addition and multiplication modulo 3. This means that after adding or multiplying numbers, if
the result is greater than or equal to 3, then we set it equal to its remainder
when divided by 3. For instance 2 + 2 = 4 which has a remainder of 1 when
divided by 3. Fill in the addition and multiplication tables:
+ 0 1 2
· 0 1 2
0
0
1
1
2
1
2
Now show that Z/3 is a field.
(4) Fill in the addition and multiplication tables for Z/4 = {0, 1, 2, 3}
+ 0 1 2 3
· 0 1 2
0
0
1
1
2
2
3
3
Is this a field? Why or why not?
(5) Find all the roots, including complex roots, of x3 − 1.
(6) Recall the correspondence between the
complex
numbers C and the Eua
b
clidean plane R2 . Now find a matrix
representing multiplication
c d
a b
x
by i. This means that
should correspond to i(x + iy).
c d
y 

1 2
1 2 3 4


3 4
(7) Evaluate the real matrix multiplication
.
5 6 7 8
5 6
0
i
2i
(8) Evaluate the complex matrix multiplication
.
−i 1 + i
3i
1