Results 1 to 3 of 3

Math Help - fifth root of unity

  1. #1
    Newbie
    Joined
    Jan 2011
    Posts
    12

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    The fifth roots of unity are simply e^{\frac{2k\pi i}{5} for k=0,1,2,3,4.

    In general, the nth roots of unity are e^{\frac{2k\pi i}{n} for k=0,1,...,n-1.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by gpenguin View Post
    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 (x^2+\frac12x+1)^2 = x^4+x^3+\frac94x^2 +x+1 = (x^4+x^3+x^2 +x+1) + \frac54x^2 and therefore x^4+x^3+x^2 +x+1 = (x^2+\frac12x+1)^2 - \bigl(\frac{\sqrt5}2x\bigr)^2. Now use the difference of two squares to factorise that as (x^2+\frac12x+1+\frac{\sqrt5}2x) (x^2+\frac12x+1-\frac{\sqrt5}2x).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. primitive root of unity
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 25th 2010, 01:04 AM
  2. Complex root of unity
    Posted in the Calculus Forum
    Replies: 4
    Last Post: December 29th 2009, 01:00 AM
  3. Sum of powers of unity root
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: August 15th 2008, 10:07 AM
  4. Primitive pth root of unity of a prime p
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 23rd 2008, 06:38 AM
  5. Primitive root of unity problem
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: March 28th 2008, 05:38 AM

Search Tags


/mathhelpforum @mathhelpforum