RESULTANTS OF CYCLOTOMIC POLYNOMIALS TOM M. APOSTOL

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1 RESULTANTS OF CYCLOTOMIC POLYNOMIALS TOM M. APOSTOL 1. Introduction. The cyclotoic polynoial Fn(x) of order ras: 1 is the priary polynoial whose roots are the priitive rath roots of unity, (1.1) Fn(x) = fl' (x - e*"*'*), where the ' indicates that the index k runs through integers relatively prie to n. The degree of Fn(x) is <f>(n), Euler's totient. This paper deterines the resultant p{f, Fn) of any two cyclotoic polynoials F and F. Explicit forulas are given which show that if p^n the resultant is either 1, 2, or a prie power. For the case wz>ra>l the results agree with a forula derived by Diederichsen [3, Hilfssatz 2] in a paper on group representations (see Theore 4 below). Our proof is different fro and soewhat sipler than that of Diederichsen; it is based on the following lea on decopositions of reduced residue systes which the author has recently used to relate Gauss sus and priitive characters [l, Lea 6]. Lea. Let Sk denote a reduced residue syste odulo k, and let d be a divisor of k. Then 5* is the union of 4>(k)/4>{d) disjoint sets, each of which is a reduced residue syste odulo d. We also ake use of the following well-known forulas for cyclotoic polynoials [2, p. 31 ], [4, Chapter 8]: (i.2) and X" -1 = n p*(*) FJ1) = 0 if n = 1, (1.3) = p if ra = p", p prie, a ^ 1, Property (1.3) is an easy consequence of (1.2) and the relation Fp.(x) = Fp(y), y = xv"~\ p prie, a>l (see [4, p. 67]). We also use the fact that each Fn(x) has integer coefficients [4, p. 61 ]. Received by the editors June 23,

2 458 T. M. APOSTOL [March 2. Properties of resultants. Given two polynoials A and B, say n A(x) = ^akxk and B{x) = ^ bkxk, t=0 fc=0 their resultant p(a, B) is denned to be the deterinant O-n O-n-l 0n-2 " ' ' «1 «0 #n On-1 02 Ol Oo w rows b b-i b-2 bi bo b b-i bz bi b0 n rows b b~i bi bo the reaining entries being equal to zero. This forula shows that p(a, B) is a polynoial in the a, and h, with integer coefficients. In particular, if all the a< and bj are integers then p(a, B) is also an integer. Hence p(fn, F) is an integer for any two cyclotoic polynoials. If A and B are expressed in ters of their zeros, say n A(x) = an IX 0 xk), B{x) = 6 II (*?>)> then the resultant can also be expressed as a product, (2.D n pu, s) = <»r iin (**-*) t-i y i A proof of (2.1) is given in [5]. This forula iplies the ultiplicative property (2.2) P(A,BC)=P{A,B)p(A,C) for any polynoials A, B, C; the syetry property (2.3) p{a,b) = (-l)-»p(s, A); and the factorization forula (2.4) P(A,B) =blfla(yj). y-i

3 i97o] RESULTANTS OF CYCLOTOMIC POLYNOMIALS 459 Two polynoials A and B have a root in coon if and only if p(a,b) =0. In particular, p(f, Fn) =0if and only if = n. 3. The resultant of Fx and F. Applying equation (2.1) to the cyclotoic polynoials Fi and F, where >\, we find (3.1) p{f1, F) = II' (1 - e2"'*"") = FM)- Using (1.3) we obtain Theore l.if>lwehave p(fi, F) = p if = p", p prie, a ^ 1, It should be noted that p(f, F{) = (-l)*^p(fu F), so p(f2, Fi) = -p(fu F,)=-2 and p(f, Fi) =p(fu F ) if >l. 4. A product forula for p(f, Fn) when >n> 1. The restriction >n>\ is not serious because P(F, Fn) = (-l)*w*wp(f., F) = P(F, Fro). We use the lea of 1 to prove Theore 2. If >n>\ we have (4.1) p(f> F ) = II p^'^^i*^, d,p where the product is extended over those divisors d of n and those pries p such that /(, d) =pa for soe a'sz 1. Proof. Using (1.2) and the ultiplicative property (2.2) we obtain (4.2) p(f,x»-l) = Jlp(F,Fd), Since»ra>ra>l each factor in (4.2) is nonzero and we can apply the Mobius inversion forula to obtain (4.3) p(f, Fn) = II P(F, x* - 1)**«>. Using the syetry property (2.3) and equation (2.4) we find p(f, x* - 1) = (-l)rf*(»)p(^ - 1, F) = II' (e2"'m/- 1). it-i In the exponential we write

4 460 T. M. APOSTOL (March kd kd/s ( d\ =, where 8= (,d), I, ) = 1. /8 \8 8 / By the lea, as k runs through a reduced residue syste odulo the product kd/s runs through a reduced residue syste odulo /8 with each residue appearing exactly <p()/(p(/8) ties. Therefore / JS \ <f,()/<f>(.l8) p(f, x* - 1) - J IT (e2">/(""8) - 1) Using (1.3) to evaluate = F/a(l)* (»>/* <»/ >..F/a(l) we find p(f, xd 1) = *()/*(/s) if ot/5 = a for soe prie p, Substituting this in (4.3) we obtain Theore Evaluation of p(f, Fn) for >n>\. We consider two cases, (, n) = 1 and (, n)>\. If (w, w) = 1 then (, d) = l for every divisor d of w, so the product in Theore 2 is epty unless is a prie power. If»w = >a the product in Theore 2 becoes n ^(n/<i) = i din since ^<ji p(n/d) =0 for w> 1. In other words, we have proved: Theore 3. If >n>l and (, n) = 1, /Aew p(f, Fn) = 1. Next we consider the case in which w and w are not relatively prie. In this case we obtain Theore 4. If >n>\ and (, n)>\, then p(f, Fn) = ^*(n) if/n is a power of a prie p, Proof. We replace d by n/d in the product (4.1) and rewrite it in the for (5.1) p(f, Fn) = n ^w>*<"» <» > d,p where the product is extended over those divisors d of n and those pries p such that /{, n/d) =p". Because of the Mobius function we need consider only square-free divisors d. We write = k', n = kn', where k = (, n) and (r,»') =!. Then

5 1970] RESULTANTS OF CYCLOTOMIC POLYNOMIALS 461 k' k'n' 'd 'd (, n/d) (k', kn'/d) kn'('d, n')/d ('d, n') 8 where 8 = ('d, n'). Since (', n') = l we have (5, ') = l so 8\d. Therefore 'd/8 is a ultiple of '. For this to be a prie power, both ' and d/8 ust be powers of the sae prie. If ' is not a prie power the product in (5.1) is epty and p(f, Fn) = 1. Assue, then, that ' is a prie power, say ' = pa. We seek those divisors d of ra for which d/8 is a power of the sae prie, say d/8=pe. This iplies d = 8ps, j3 0. Now ra = kn' and d ra so 5/>" &ra', hence pb\ kn'. But (/>, «') = 1 since (', ra') = 1, sop0\ k. Therefore we can write k = p~>k', where (p, k') = 1. We now have 'd n = kn' = pik'ri, = k' = p^k', -=-= pa+l>. (, n/d) 8 We also have d = 8pe. Since d is square-free this requires 0 = 0 or j8 = l, so each rf has the for 5 or 5, where (/>, 5) = 1. Now d\ n so d pyk'n'. If we let d' range through all the square-free divisors of k'n' we see that the possible values of d are all the divisors d' (these correspond to (8 = 0) plus all products of the for pd' (these correspond to j3 = 1). The contribution to the product in (5.1) fro each divisor d' is p raised to the power p.(d')(p()/4>(pa). The contribution fro each divisor pd' is p raised to the power n(d')(j>()/<f)(pa+l). But we have and, siilarly, Therefore <j,() <t>(p +y)4>(k') ph(p*w) <kp ) <s>(pa) <t>(p*) - = - = - = i>"i<b(k ) 4>()/4>(pa+1) = p*-w). <K)/4>(p ) - <j,()/<t>(p«+l) = (py - p->-l)4>(k') = *f>w) = *(*). Therefore the contribution to the product fro each pair of divisors d = d' and d = pd' is />ew'>*(*>. We do not alter the product if we also include as factors the sae power of p taken over the nonsquare-free divisors of k'n'. Therefore we obtain p(f, Fa) = If />"(d'>*(t' = {?*<»}', d'\k>n'

6 462 T. M. APOSTOL where I = Z (/) = 1 if k'n' = 1, d'lk'n' = 0 if k'n' > 1. This shows that p(f, F ) = 1 unless k'n' = 1, in which case p(f, F ) = p*ci). But k'n' = l iplies k' = n'=l and this iplies n = k = py, = npa. Therefore the resultant is equal to 1 unless /n = pa, in which case p(f, Fn) =p*'"). This copletes the proof of Theore 4. References 1. To M. Apostol, Euler's <f>-function and separable Gauss sus, Proc. Aer. Math. Soc. 24 (1970), L. E. Dickson, H. H. Mitchell, H. S. Vandiver and G. E. Wahlin, Algebraic nubers, Bulletin of the National Research Council, vol. 5, part 3, no. 28, National Acadey of Sciences, Fritz-Erdann Diederichsen, Uber die Ausreduktion ganzzahliger Gruppendarstellungen bei arithetischer Aquivalenz, Abh. Math. Se. Hanischen Univ. 13 (1940), MR 2, Hans Radeacher, Lectures on eleentary nuber theory, Blaisdell, Waltha, Mass., MR 30 # B. L. van der Waerden, Moderne algebra. Vol. I, 2nd rev. ed., Springer, Berlin, 1937; English transl., Ungar, New York, MR 2, 120; MR 10, 587. California Institute of Technology

Bernoulli Numbers. Junior Number Theory Seminar University of Texas at Austin September 6th, 2005 Matilde N. Lalín. m 1 ( ) m + 1 k. B m.

Bernoulli Numbers. Junior Number Theory Seminar University of Texas at Austin September 6th, 2005 Matilde N. Lalín. m 1 ( ) m + 1 k. B m. Bernoulli Nubers Junior Nuber Theory Seinar University of Texas at Austin Septeber 6th, 5 Matilde N. Lalín I will ostly follow []. Definition and soe identities Definition 1 Bernoulli nubers are defined