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CYCLOTOMIC POLYNOMIALS 1. the Derivative and Repeated Factors

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CYCLOTOMIC POLYNOMIALS 1. the Derivative and Repeated Factors CYCLOTOMIC POLYNOMIALS 1. The Derivative and Repeated Factors The usual definition of the derivative in calculus involves the nonalgebraic notion of limit that requires a field such as R or C (or others) where limits are sensible for analytic reasons. However, polynomial differentiation is an algebraic notion over any field. One can simply define the polynomial derivative to be as it is from calculus, n ! n−1 X i X i D aiX = (i + 1)ai+1X : i=0 i=0 Alternatively (and working a bit casually here), one can introduce a second inde- terminate T and observe that the polynomial f(X + T ) − f(X) 2 k[X][T ] is satisfied by T = 0, and so it takes the form f(X + T ) − f(X) = T g(X)(T ); g 2 k[X][T ]: The derivative of f is then defined as the T -constant term of g(X)(T ), f 0(X) = g(X)(0) 2 k[X]: The familiar facts that the derivative is linear, i.e., (f + cg)0 = f 0 + cg0, and that the derivative satisfies the product rule, i.e., (fg)0 = fg0 + f 0g, can be rederived from this purely algebraic definition of the derivative. In fact, yet a third approach is to define X0 = 1 and then stipulate that differentiation is the unique extension of this rule that is linear and satisfies the product rule. (For example, 1 = X0 = (X · 1)0 = X · 10 + X0 · 1 = X · 10 + 1, so 10 = 0.) Proposition 1.1. Let k be a field, and let f 2 k[X] be a polynomial. Then: If gcd(f; f 0) = 1 then f has no repeated factors. Proof. To establish the contrapositive, suppose that f = g2h in k[X]. The product rule gives f 0 = 2gg0h + g2h0 = g(2g0h + gh0): 0 0 0 Thus g j f , and so g j gcd(f; f ), and so gcd(f; f ) 6= 1. 2. Definition of the Cyclotomic Polynomials Working in Z[X] we want to define cyclotomic polynomials Φn; n 2 Z≥1: Provisionally, define Xn − 1 Φn(X) = Q ; n ≥ 1: djn Φd(X) d<n 1 2 CYCLOTOMIC POLYNOMIALS Here it is understood that for n = 1 the denominator is the empty product, i.e., Φ1(X) = X − 1: Certainly each Φn lies in the quotient field Z(X) of Z[X] but we will show con- siderably more. In fact, always Φn lies in Z[X] and is irreducible. For example, n Q repeatedly using the identity X − 1 = djn Φd(X) in various ways, X2 − 1 Φ (X) = = X + 1 = −Φ (−X); 2 X − 1 1 Xp − 1 Φ (X) = = Xp−1 + ··· + X + 1; p prime; p X − 1 pe X − 1 pe−1 Φ e (X) = = Φ (X ); p prime; p Xpe−1 − 1 p 2dp 2d−1p X − 1 X + 1 2d−1 Φ d (X) = = = Φ (−X ); p > 2 prime; 2 p 2d−1p 2d−1 p (X − 1)Φ2d (X) X + 1 d e X2 p − 1 Φ d e (X) = 2 p 2d−1pe (X − 1)Φ2d (X)Φ2dp(X) ··· Φ2dpe−1 (X) d−1 e X2 p + 1 = (induction on e) 2d−1 2d−1 2d−1 −Φ1(−X )Φp(−X ) ··· Φpe−1 (−X ) 2d−1pe X + 1 d−1 e−1 = = Φ (−X2 p ); p > 2 prime; X2d−1pe−1 + 1 p (X15 − 1) X10 + X5 + 1 Φ15(X) = 5 = 2 (X − 1)Φ3(X) X + X + 1 = X8 − X7 + X5 − X4 + X3 − X + 1; Φ2m(X) = Φm(−X); m odd; X21 − 1 X14 + X7 + 1 Φ21(X) = 7 = 2 (X − 1)Φ3(X) X + X + 1 = X12 − X11 + X9 − X8 + X6 − X4 + X3 − X + 1: These identities give Φn for all n ≤ 32. They also suggest that all cyclotomic polynomials Φn for n > 1 have constant term 1, and this is readily shown by induction on n. As an aside, invoking some ideas from outside this course, the relation n Y X − 1 = Φd(X); n 2 Z≥1 djn gives in consequence a closed form expression for the cyclotomic polynomials, Y d µ(n=d) Φn(X) = (X − 1) ; n 2 Z≥1; djn in which µ is the M¨obiusfunction of elementary number theory, g ( g Y (−1) if ei = 1 for each i; µ( pei ) = i 0 if e ≥ 2 for some i: i=1 i Here it is understood that terms p0 are omitted from the product in the previous display, and that we view 1 as the empty product of prime powers, so that the CYCLOTOMIC POLYNOMIALS 3 0 formula gives µ(1) = (−1) = 1. The formula for the cyclotomic polynomial Φn(X) as a product of polynomials Xd − 1 and their reciprocals, in consequence of the n formula for the polynomial X − 1 as a product of cyclotomic polynomials Φd(X), is an instance of M¨obiusinversion. n We begin with two observations about the polynomial X − 1 where n 2 Z≥1. n • X − 1 has no repeated factors in Z[X] for any n 2 Z≥1. Indeed, working in the larger ring Q[X] we have n n 0 n n−1 × (X − 1) − (1=n)X · (X − 1) = (X − 1) − (1=n)X · nX = −1 2 Q ; so that gcd(Xn − 1; (Xn − 1)0) = 1. As explained above, Xn − 1 therefore has no repeated factors in Q[X], much less in Z[X]. For future reference, we note that this argument also works in (Z=pZ)[X] where p is prime, so long as p - n. • For any n; m 2 Z≥1, gcd(Xn − 1;Xm − 1) = Xgcd(n;m) − 1: Indeed, taking n > m, the calculation Xn − 1 − Xn−m(Xm − 1) = Xn−m − 1 shows that hXn − 1;Xm − 1i = hXn−m − 1;Xm − 1i; and, letting n = qm + r where 0 ≤ r < n and repeating the calculation q times, hXn − 1;Xm − 1i = hXr − 1;Xm − 1i: That is, carrying out the Euclidean algorithm on Xn − 1 and Xm − 1 in Q(X) slowly carries out the Euclidean algorithm on n and m in Z. (If the Euclidean algorithm on n and m takes k division-steps, with quotients n m q1 through qk, then the Euclidean algorithm on X − 1 and X − 1 takes q1 + ··· + qk steps.) The result follows. Recall the definition Xn − 1 Φn(X) = Q ; n ≥ 1: djn Φd(X) d<n Note that • Φ1 2 Z[X], and • (vacuously) no Φk and Φm for distinct k; m < 1 share a factor. The two bullets are the base case of an induction argument. Fix some n ≥ 1 and assume that • all Φm for m ≤ n lie in Z[X], and • no Φk and Φm for distinct k; m < n share a factor. Let m < n, and let k = gcd(m; n) ≤ m < n. Any common factor of Φm and Φn is a common factor of Xm − 1 and Xn − 1, hence a factor of Xk − 1. However, the calculation Xn − 1 Xn − 1 Xn − 1 Φ (X) = = n Q Q k djn Φd(X) djn Φd(X) X − 1 d<n d≤k 4 CYCLOTOMIC POLYNOMIALS k shows that no factor of Φn is a factor of X − 1. In sum, the strict inequality in the second bullet weakens: no Φk and Φm for distinct k; m ≤ n share a factor. Equivalently, no Φk and Φm for distinct k; m < n + 1 share a factor. d Now, for each proper divisor d of n + 1, each factor of Φd is a factor of X − 1 in Z[X], hence a factor of Xn+1 − 1 in Z[X]. And the product Q djn+1 Φd(X) d<n+1 n+1 contains no repeat factors of X − 1. Thus Φn+1 2 Z[X]. This completes the induction step: • all Φm for m ≤ n + 1 lie in Z[X], and • no Φk and Φm for distinct k; m < n + 1 share a factor. By induction, Φn 2 Z[X] for all n 2 Z≥1, and no two Φn share a factor. P The totient function identity djn φ(d) = n quickly combines with the formula Q n djn Φd(X) = X − 1 and a little induction to show that the degree of the cyclo- tomic polynomial is the totient function of its index, deg(Φn) = φ(n); n ≥ 1: If we view the integers Z as a subring of the complex number field C then the nth cyclotomic polynomial factors as Y e 2πi=n Φn(X) = (X − ζn); where ζn = e : 0≤e<n gcd(e;n)=1 Indeed, the formula produces the monic polynomial in C[X] whose roots are the complex numbers z such that zn = 1 but zm 6= 1 for all 1 ≤ m < n. Indeed, the word cyclotomic literally refers to dividing the circle. The previous paragraph notwithstanding, we do not want to think of the cyclo- tomic polynomials innately in complex terms, because the integers are also compati- ble with other algebraic structures that are incompatible with the complex numbers in turn. Rather, we want to think of the roots of Φn as the elements having order exactly n in whatever environment is suitable. In the next example, the environ- ment is Z=pZ where p is a prime that does not divide n. Earlier we observed that our arguments apply in this setting as well. 3. Application: An Infinite Congruence Class of Primes Theorem 3.1. Let n > 1 be a positive integer. There exist infinitely many primes p such that p = 1 mod n.
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