1. ## Primitive roots

I am trying to show that the primitive fifth root of unity e^(2(pi)i/5)=sqrt5-1/4 + i (sqrt (5)+(sqrt5)/8. I have solved (sqrt (5)-sqrt5/8 but not sure really how to complete the algebra?

I am trying to show that the primitive fifth root of unity
We know the n>1-th roots of unity that are of the form e^{m*pi*i/n} where m is a positive integer and gcd(m,n)=1 will be a generator for the group of all roots of unity.
Since 5 is a prime there are exactly 4 primitive roots of unities (because those the ones that form generators):
e^{pi*i/5},e^{2pi*i/5},e^{3pi*i/6},e^{4pi*i/5}

3. ## Sorry but

I am trying to show that the primitive fifth root of unity e^(2(pi)i/5)=sqrt5-1/4 + i (sqrt (5)+(sqrt5)/8. I have solved (sqrt (5)-sqrt5/8 but not sure really how to complete the algebra?
I'm sorry I can't answer your question but I bet you can answer mine. What is a primitive root?

More careful placement of parentheses would have helped.

And I still don't understand what you did . . .

I am trying to show that the primitive fifth root of unity is:
. . . . . - - - . . . . . ._ . . . . . . . . _________
. . e^{2πi/5} .= .[√5 - 1]/4 + i·√(5 + √5)/8

. . . . . . . . . . . ________
I have solved √(5 - √5)/8 . ? . and what do you mean by "solved"?

From DeMoivre's Formula, we already know that:

. . e^(2πi/5) .= .cos(2π/5) + i·sin(2π/5)

So we need to find the values of: .cos(72°) and sin(72°)

Can you do that?

5. Originally Posted by exweedfarmer
I'm sorry I can't answer your question but I bet you can answer mine. What is a primitive root?
It is not an elementary answer but here it is.

Given a finite field F. It can be shown that the multiplicative group is in fact cyclic! Thus, it has generators. Those generators are primitive roots.