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Roots of Unity, Roots of Complex Numbers, and Cyclotomic Polynomials

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Roots of Unity, Roots of Complex Numbers, and Cyclotomic Polynomials 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).
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