Section notes, October 13, 1999.

- What is a complex 'n'th root of unity?

  One answer is "any complex number that when taken to the 'n'th power gives
  1."

  * Example: 1 is a 'n'th root of unity for any n.

  * Example: solve the equation x^2=2.  High school algebra tells us the
    answer is x = +/- sqrt(2) = (+/- 1) * sqrt(2).  How about x^3 = 2?  In
    algebra 1 they told us that since 3 is odd, the only answer will be
    cube-root(2).  However, the complex number cube-root(2) * exp(i * 2*pi/3)
    is also a solution.

- Think of the complex 'n'th roots of unity as the solutions to the
  equation x^n = 1.  The fundamental theorem of algebra tells us that
  there are n solutions to this equation, but not necessarily distinct
  and not necessarily real.  Common sense says that there are at most
  two real solutions: 1, and if n is even, -1.  Figure 32.2 in the
  book confirms this: we have to put n equally spaced vectors in a
  circle starting with one pointing to the right.  By symmetry, we can
  argue that if n is even, just as many of them will be above the
  horizontal axis as below it, so for balance we need a vector
  pointing *on* the line to the left (which is -1).  If n is odd, we
  need there to still be as many above as below, but there is no
  "leftover" vector that needs to be on the line.

- Why are they equally spaced?  Well, the vector exp(i * 2*pi/n) is
  definitely an 'n'th root of unity.  What about exp(i * b*2*pi/n), where b
  is an integer between 2 and n?  Well,

     exp(i * b*2*pi/n) = exp(i * 2*pi/n)^b.

  Then by taking it to the nth power, we get,

     ( exp(i * 2*pi/n)^b )^n = exp(i * 2*pi/n)^(n*b) =
     
     ( exp(i * 2*pi/n)^n )^b.

  But the thing inside the parentheses at the end is just exp(i * 2*pi/n)
  raised to the 'n'th power, which we already know is 1.  So the whole
  thing is ( 1 )^b = 1.

- So we have these vectors with angles b*2*pi/n, where j ranges from 1 to n.
  Another way to write these is as exp(i * b*2*pi/n).
  (Aside: e^i might be a source of worry, since i is not a real number, so
   we might ask what it "means" to write down that formula.  The answer is
   that we are allowed to define it as whatever we want -- but hopefully it
   would still make sense.  One way to define it is just as you did in the
   previous homework: with the Taylor series expansion of e^x, for an
   arbitrary variable x.  In fact, one can even define e^A when A is a
   matrix.)

- Book terminology: w_n is called the "principal" 'n'th root of unity,
  defined as the root with the smallest argument (angle in the complex
  plane): exp(i * 2*pi/n), and all others are defined as powers of it.
  However, as noted in lecture, in general many 'n'th roots of unity
  satisfy the property that all of the other 'n'th roots are powers of
  it.  For example, (w_5)^2 = exp(i * 2*pi*2/5) is one, while (w_6)^2
  = exp(i * 2*pi*2/6) is not.  (As an exercise, can you find values of
  n where _all_ 'n'th roots have this property?)

- Practice: lemma 32.6: "For any integer n>=1 and nonnegative integer k
  not divisible by n, sum(j=0, n-1, ((w_n)^i)^j) = 0.

  Proof:











- Dividing polynomials.

  * Example: the one on page 798:

    (3x^3 + x^2 - 3x + 1) mod (x^2 + x + 2)











  * Example: verifying the statement on page 799: "note that Q_{i,j}(x)
    has degree bound at most j-i"