1. ## Complex numbers

How do you show that cube roots of unity (the set of complex numbers which satisfy z^3=1) for a group?

2. Originally Posted by chmaths
How do you show that cube roots of unity (the set of complex numbers which satisfy z^3=1) for a group?
What is a group?
The operation in this group is multiplication.
So show that axioms for a group are satisfied.

3. 1. associativity (of multiplication) holds for any subset of the complex numbers (because the complex numbers form a ring).

2. what is the multiplicative identity of C? is it a cube root of 1? (don't over-think this)

3. how many cube roots of 1 are there? you can use the fundamental theorem of algebra. (if z^3 = 1, then z is a root of x^3 - 1.

how many roots can a cubic polynomial have at most?). besides the "obvious" root of x^3 - 1, (which being a difference of two cubes,

you should at least be able to factor a little bit) what are the "other roots"? and what is the product of these "other roots"?

how does this tell you we have inverses?