- #1
auru
- 10
- 0
Homework Statement
I am required to find the units of ℤ8.
Homework Equations
I have that
##\bar{a}## = [a]n = { a + kn, k ∈ ℤ }
##u## ∈ ℤn is a unit if ##u## divides ##\bar{1}##.
The Attempt at a Solution
I'm not sure how to go about this. My lecturer wrote out the multiplication table for ℤ8 and simply noted that by inspection of the table, the units are: ##\bar{1}##, ##\bar{3}##, ##\bar{5}##, ##\bar{7}##.
So I have the multiplication table
ℤ8 ##\bar{0}##, ##\bar{1}##, ##\bar{2}##, ##\bar{3}##, ##\bar{4}##, ##\bar{5}##, ##\bar{6}##, ##\bar{7}##,
##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##, ##\bar{0}##,
##\bar{1}##, ##\bar{0}##, ##\bar{1}##, ##\bar{2}##, ##\bar{3}##, ##\bar{4}##, ##\bar{5}##, ##\bar{6}##, ##\bar{7}##,
##\bar{2}##, ##\bar{0}##, ##\bar{2}##, ##\bar{4}##, ##\bar{6}##, ##\bar{0}##, ##\bar{2}##, ##\bar{4}##, ##\bar{6}##,
##\bar{3}##, ##\bar{0}##, ##\bar{3}##, ##\bar{6}##, ##\bar{1}##, ##\bar{4}##, ##\bar{7}##, ##\bar{2}##, ##\bar{5}##,
##\bar{4}##, ##\bar{0}##, ##\bar{4}##, ##\bar{0}##, ##\bar{4}##, ##\bar{0}##, ##\bar{4}##, ##\bar{0}##, ##\bar{4}##,
##\bar{5}##, ##\bar{0}##, ##\bar{5}##, ##\bar{2}##, ##\bar{7}##, ##\bar{4}##, ##\bar{1}##, ##\bar{6}##, ##\bar{3}##,
##\bar{6}##, ##\bar{0}##, ##\bar{6}##, ##\bar{4}##, ##\bar{2}##, ##\bar{0}##, ##\bar{6}##, ##\bar{4}##, ##\bar{2}##,
##\bar{7}##, ##\bar{0}##, ##\bar{7}##, ##\bar{6}##, ##\bar{5}##, ##\bar{4}##, ##\bar{3}##, ##\bar{2}##, ##\bar{1}##,
By inspection of the table, I see that ##\bar{1}##, ##\bar{3}##, ##\bar{5}##, ##\bar{7}## all yield rows where each product is unique, indicating that they are the units of ℤ8. However, I'd like to know of a more concrete way of finding the units of ℤn, if that is possible.