Roots of Unity, Roots of Complex Numbers, and Cyclotomic Polynomials Gary D. Knott Civilized Software Inc. 12109 Heritage Park Circle Silver Spring MD 20906 email:[email protected] URL:www.civilized.com November 14, 2017 0.1 Complex Numbers The set of complex numbers is the set α + iβ α,β where i denotes the “imaginary” value C { | ∈ R} √ 1; α + iβ is a “complex” of a real part α and an imaginary part iβ; the imaginary part is 0 in − the case where β = 0. The real numbers are a subset of the complex numbers, and we generally R leave out the real or imaginary part in writing a complex number when that part is 0. Arithmetic with complex numbers follows the usual rules, including i−1 = i and i2 = 1, and is a field. − − C There are some special features that make complex number arithmetic different than computation with real numbers. Given a complex number z = α+iβ, we define Re(z)= α and Im(z)= β. Let z and w be two complex numbers. The sum z + w is the complex number Re(z)+Re(w)+ i(Im(z)+ Im(w)) and the product zw is the complex number Re(z)Re(w) Im(z)Im(w)+ i(Re(z)Im(w)+ − Im(z)Re(w)). This is consistent with i2 = 1. The conjugate of z is the complex number − z∗ = α iβ = Re(z) i Im(z). (Often the conjugate of z is writtenz ¯.) The magnitude of z is the − − non-negative real number (zz∗)1/2 denoted by z . (The magnitude of a complex number z is also | | called the modulus of z.) Note Re(z)=(z + z∗)/2 and Im(z)=(z z∗)/(2i). − Exercise 0.1: Let z . Show that z2 = z 2. ∈C | | | | Exercise 0.2: Let z,w . Show that z∗∗ = z, (z + w)∗ = z∗ + w∗, and (zw)∗ = z∗w∗. This ∈C means the self-inverse mapping z z∗ is an automorphism of ; when we adjoin i to (and → C R “stir”) to form , we can’t tell the difference between choosing i = √ 1 versus i = √ 1 from C − − − “inside” . C Exercise 0.3: Let z = a + ib with a, b and z = 0. Show that 1/z = z∗/(z∗z). ∈R 6 Exercise 0.4: Show that for z with z = 1, z + z−1 = 2Re(z). ∈C | | The field of complex numbers has the wonderful property that any polynomial p(z)= p + p z + C 0 1 + p zn with coefficients p ,p ,...,p where p = 0 can be factored into n linear factors so ··· n 0 1 n ∈C n 6 1 2 that p(z)= p (z r )(z r ) (z r ) where r ,r ,...,r . (The values r ,...,r need not be n − 1 − 2 ··· − n 1 2 n ∈C 1 n distinct.) The values r1,...,rn are called the roots or zeros of the polynomial p, since p(rj)=0for j = 1,...,n. This fact is called the Fundamental Theorem of Algebra: every degree-n polynomial p(z) with its coefficients in has n roots in , i.e., n linear factors z r with r such that C C − j j ∈ C p(r ) = 0; it is what we mean by calling the complex numbers algebraically complete. j C The distinct roots of p are determined by the coefficients of p, and when pn = 1, the remain- ing coefficients of p are determined by the n roots of p. The coefficients of p(x)/pn are given pj in terms of elementary symmetric functions of the roots r1,...,rn; for j = 0, 1, 2,...,n, = pn ( 1)n−je (r ,r ,...,r ) where − n−j,n 1 2 n e (x ,x ,...,x ) := x x x k,n 1 2 n j1 j2 ··· jk 1≤j1<jX2<···<jk≤n for 0 k n + with e (x ,x ,...,x ) = 1 and e (x ,x ,...,x ) = 0 for k>n. The ≤ ≤ ∈ Z 0,n 1 2 n k,n 1 2 n multivariate polynomial functions ek,n(x1,x2,...,xn) are called the elementary symmetric functions because ek,n is invariant to any permutation of the arguments x1,x2,...,xn. Exercise 0.5: Let p(x) = p + p x + + p xn = p (x r )(x r ) (x r ) with 0 1 ··· n n − 1 − 2 ··· − n p ,p ,...,p ,r , ,r . Define p∗(x) = p∗ + p∗x + + p∗ xn. Show that the polyno- 0 1 n 1 ··· n ∈ C 0 1 ··· n mial p(x)p∗(x) has real coefficients. Also show that the roots of p∗ are the conjugates of the roots of p. Thus if the polynomial p has only real coefficients, then if r is a root of p, r∗ is also a root of p. Hint: the polynomials p and p∗ are functions defined on . C Euler’s formula eiθ = cos(θ)+ i sin(θ) allows us to write any non-zero complex number z in polar form z = z ei arg(z) where arg(z) := atan2(Im(z)/ z , Re(z)/ z ) for z = 0. Treating z as a point | | · | | | | 6 (Re(z), Im(z)) in 2, arg(z) is the angle in ( π,π] between the x-axis and the vector (Re(z), Im(z)); R − this angle is negative if Im(z) < 0 and non-negative if Im(z) 0. We define arg(z) := π/2 for ≥ z = 0. When we look at the components of points in 2 as being the real values comprising R the real and imaginary parts of complex numbers, we see that the points of 2 are in one-to-one R correspondence with , and in this context, we call 2 the complex plane. C R Exercise 0.6: Let z . Show that z = z cos(arg(z))+i z sin(arg(z)). Hint: draw a vector ∈C | | | | in the complex plane corresponding to z. Exercise 0.7: Show that eiπ = e−iπ = 1. Here again, we can’t tell the difference be- − tween √ 1 and √ 1, or equivalently, between clockwise rotations in righthanded 2-space and − − − counter-clockwise rotations in lefthanded 2-space. Let the complex numbers z and w be given in polar form as z = z eiθ and w = w eiφ. Then | | | | the product zw in polar form is zw = z eiθ w eiφ = z w ei(θ + φ). This exhibits the fact that, | | | | | || | looking at z, w, and zw as vectors in the right-handed complex plane, zw is the vector z rotated counter-clockwise arg(w) radians and scaled in length by w . | | Exercise 0.8: Look-up the development of Euler’s formula based on the power-series for ex, sin(θ), and cos(θ). 3 Because angle(x,y) (without restricting qualification), is multi-valued, (i.e., the angle between two non-zero 2-vectors is not a unique value but instead can be any value in the set of real numbers R arccos((x,y)/( x y )) + k2π k ), arg is more properly taken to be a multi-valued function; { | || | | ∈ Z} for z = 0, we may take arg(z) to be any of the values atan2(Im(z)/ z , Re(z)/ z ) + k2π with 6 | | | | k . The definition we gave before is the principal value arg function. We may have multi-valued ∈Z expressions involving complex numbers when Euler’s formula is involved in the definition of the expression. (Note cos(arg(z)) is not a multi-valued function, but arg(z) cos(arg(z)) is a multi- · valued function.) Generally however, in the sequel we shall take arg(z) as denoting its principal value. Exercise 0.9: Show that the principal value of arg(1) is 0. Note ei2π = 1. This is consistent with the fact that for r +, really, reiθ = r(cos(θ + k2π)+ ∈ R i sin(θ + k2π)) for k . We generally take k = 0 to obtain the principal value of arg(reiθ) ∈ Z in ( π,π]. Since ew + i2π = 1, ew = ew + i2π, and we see that the complex function ew is a − periodic function with the period i2π. This far-reaching fact leads to a large body of intricate relationships and requires us to deal with multi-valued functions. Of course, multi-valued functions are not functions – at least not functions producing single complex numbers as output – they are specifications of sets. But by suitably restricting the range of such “functions” to so-called principal values, we can reduce a multi-valued function to a proper function, generally with the loss of continuity along a boundary. (Even in real-number computation, multi-valued functions arise when square-roots of positive values occur. When k-th roots of positive values occur with i(2π/k) i2(2π/k) i(k 1)(2π/k) k > 2, the complex roots of unity e , e , ..., e − , 1 slip-in.) (Generally computations with real numbers are implicitly restricted to principal value computations, so that, for example, when we write √3 2 we mean the positive real cube-root of 2.) Powers and logarithms may be defined for complex numbers. First, for z 0 with z = aeiθ ∈C−{ } where a = z and π<θ π, we define log(z) to be the complex number log(a)+ iθ. This is | | − ≤ consistent with log(aeiθ)= log(a)+ log(eiθ) and log(eiθ)= iθ. Now let w 0 be given as w = beiφ where b = w and 0 φ < 2π. Then for z = 0, zw is ∈C−{ } | | ≤ 6 defined as ew log(z). In particular, ew = eRe(w) ei Im(w).