Thread: fifth root of unity

apologies if this is in the wrong place, also this is probably the first of many posts on the roots of unity.

Question:
show x^5-1 = (x-1)(x^2 + ((1+sqrt5)/2)x + 1)(x^2 + ((1-sqrt5)/2)x + 1)

My soln so far:
x^5 -1 = (x-1)(x^4 +x^3+x^2+x+1)


z5 = (x-z5)(x-z5^2)(x-z5^3)(x-z5^4)(x-z5^5)
(zeta 5)

z5= e^ ((2 pi i)/5) = cos(2pi/5)+isin(2pi/5)

no idea what to do.

The fifth roots of unity are simply for .

In general, the nth roots of unity are for .

Originally Posted by gpenguin

apologies if this is in the wrong place, also this is probably the first of many posts on the roots of unity.

Question:
show x^5-1 = (x-1)(x^2 + ((1+sqrt5)/2)x + 1)(x^2 + ((1-sqrt5)/2)x + 1)

My soln so far:
x^5 -1 = (x-1)(x^4 +x^3+x^2+x+1)

That's a good start. If you now want a purely algebraic solution, notice that and therefore . Now use the difference of two squares to factorise that as .