aa r X i v : . [ m a t h . N T ] A p r On the Middle Coefficient of a Cyclotomic Polynomial
Gregory P. Dresden
The cyclotomic polynomials Φ n for n = 1 , , , . . . (familiar to every student of algebra) arethe minimal polynomials for the primitive n th roots of unity:Φ n ( x ) = Y ( k,n )=1 (cid:16) x − e πik/n (cid:17) . Clearly Φ n has degree φ ( n ), where φ signifies Euler’s totient function. These monic polyno-mials can be defined recursively as Φ ( x ) = x − Y i | n Φ i ( x ) = x n − n >
1. The firstfew are easily calculated to be x − , x + 1 , x + x + 1 , x + 1 , . . . . For these and otherbasic facts, see an algebra text such as [5].While it might appear that the coefficients of the cyclotomic polynomials are always ±
1, the presence of 2 x in Φ ( x ) shows that this is not invariably the case (and indeedis a good counterexample for those students who insist that the “law of small numbers” isuniversally valid; see [4] for further discussion). Naturally, much work has been done on thevalues of the coefficients of Φ n ( x ). One amazing fact worthy of mention is that every integerappears as a coefficient in some cyclotomic polynomial (see [1], [8]).In this article, we provide a short and elementary proof of the following result: Theorem 1.
For n ≥ the middle coefficient of Φ n ( x ) is either zero (when n is a power of ) or an odd integer. A similar result can be found in [6], where Lam and Leung directly calculate themiddle coefficient of Φ pq ( x ) for distinct primes p and q and show it to be ±
1. This hadbeen done earlier by Beiter [2] for the case of distinct odd primes. Both papers rely on thepartition of φ ( pq ) / rp + sq . In contrast, our proof uses only some very basic facts aboutminimal polynomials. We also point out that for n = pq the polynomial Φ n ( x ) could indeedhave a middle coefficient different from 1 or −
1. The first such occurence is at n = 385(giving a middle coefficient of − n = 4785, followed by − n = 7735, and 19 at n = 11305. All these values of n are square-free products of small oddprimes, which is alluded to in [8].Before proceeding with the proof of Theorem 1, we do a bit of preliminary work. Thefirst lemma establishes a useful fact about Φ n ( x ). Lemma 1. If n ≥ and is odd, then Φ n ( −
1) = 1 .Proof.
For n ≥ Y i | n,i> Φ i ( x ) = x n − x − , so (since n odd) Y i | n,i> Φ i ( −
1) = ( − n − − − . Also, Φ ( −
1) = 1. By a simple induction argument we conclude that Φ n ( −
1) = 1 whenever n ≥ ζ n to signify a primitive n th rootof unity (that is, ζ n = e πik/n for some k relatively prime to n ), and f n ( x ) to denote theminimal polynomial of ζ n + ζ − n (recall that the minimal polynomial of an algebraic complexnumber α is the monic polynomial p ( x ) in Q [ x ] of smallest degree such that p ( α ) = 0). Itis not hard to show using elementary methods (see [7]) that f n has integer coefficients andthat when n ≥ f n is half that of Φ n ( x ). In fact,Φ n ( x ) = f n ( x + x − ) · x φ ( n ) / ( n ≥ φ ( n ), and have ζ n as a root. The first few such polynomials f n (for n ≥
3) are easy to derive from (1) and read as follows: f ( x ) = x + 1 ,f ( x ) = x, f ( x ) = x + x − ,f ( x ) = x − , f ( x ) = x + x − x − ,f ( x ) = x − . From this, we see that the constant term in f n is not always ± ζ n + ζ − n is not necessarily an algebraic unit , meaning a unit in the ring of algebraic integers). However,by doing a careful comparison of the f n with the Chebyshev polynomials, Carlitz and Thomas[3] showed that when n ≥ n is not divisible by 4, the constant term in f n ( x ) is either1 or −
1. For the sake of completeness, we provide a nonelementary, but much shorter, proofof this fact.
Lemma 2. If n ≥ and n , then ζ n + ζ − n is an algebraic unit.Proof. Let m = n for n odd and m = n/ n even. Note that m is itself odd and m ≥ ζ n is a primitive m th root of unity (and thus a root of Φ m ( x )). Then ζ n +1is a root of Φ m ( x − m ( −
1) = 1 (byLemma 1). It follows that ζ n + 1 is an algebraic unit, as is ζ n . Thus, ζ n + ζ − n = ( ζ n + 1) /ζ n is likewise an algebraic unit.We are now ready to bring everything together. Proof of Theorem 1 . If n = 2 k , then Φ n ( x ) = x k − + 1, a polynomial with zero as its middlecoefficient. We proceed assuming that n is not a power of 2.Note that if ζ is a primitive 4 k th root of unity, then ζ is a primitive 2 k th root ofunity. Since φ (4 k ) = 2 φ (2 k ), we know that Φ k ( x ) = Φ k ( x ). Since the middle coefficientof Φ k ( x ) is the same as that of Φ k ( x ), we can further assume without loss of generalitythat 4 does not divide n .Now letting f n ( x ) be the minimal polynomial of ζ n + ζ − n , we know from Lemma 2that f n has constant coefficient ±
1. Thus, we can write f n ( x ) = x k + a k − x k − + · · · + a x ± k = φ ( n ) / n ( x ) = h ( x + x − ) k + a k − ( x + x − ) k − + · · · ± i · x k . (2)The middle coefficient of Φ n ( x ) is the coefficient of the x k term in (2) (recall, k = φ ( n ) / a i ( x + x − ) i in (2), plus the final ±
1. The constant term in a i ( x + x − ) i is either zero (for i odd) or a i (cid:16) ii/ (cid:17) (for i even). As a result, the middle coefficient of Φ n ( x ) is X i =2 j a i ii/ ! ± X j a j jj ! ± . (3)By a familiar identity, jj ! = j − j − ! + j − j ! = 2 j − j ! . Thus the middle coefficient of Φ n ( x ) is odd when n is not a power of 2. References [1] S. D. Adhikari, S. A. Katre, and D. Thakur, eds.,
Cyclotomic Fields and Related Topics ,Bhaskaracharya Pratishthana, Pune, 2000.[2] M. Beiter, The midterm coefficient of the cyclotomic polynomial F pq ( x ), Amer. Math.Monthly (1964) 769–770.[3] L. Carlitz and J. M. Thomas, Rational tabulated values of trigonometric functions, Amer.Math. Monthly (1962) 789–793.[4] R. K. Guy, The strong law of small numbers, Amer. Math. Monthly (1988) 697–712.[5] T. W. Hungerford, Algebra , Springer-Verlag, New York, 1980.[6] T. Y. Lam and K. H. Leung, On the cyclotomic polynomial Φ pq ( X ), Amer. Math.Monthly (1996) 562–564.[7] D. H. Lehmer, A note on trigonometric algebraic numbers,
Amer. Math. Monthly (1933) 165–166.[8] J. Suzuki, On coefficients of cyclotomic polynomials, Proc. Japan Acad. Ser. A Math.Sci. (1987) 279–280.Dr. Gregory Dresden (540) 458-8806Department of Mathematics, Robinson Hall [email protected]@wlu.edu