Solving Cyclotomic Polynomials by Radical Expressions Andreas Weber and Michael Keckeisen

Transcription

1 Solving Cyclotomic Polynomials by Radical Exressions Andreas Weber and Michael Keckeisen Abstract: We describe a Male ackage that allows the solution of cyclotomic olynomials by radical exressions. We rovide a function that is an extension of the Male solve command. The major algorithmic ingredient of the ackage is an imrovement of a method due to Gauss which gives radical exressions for roots of unity. We will give a summary for comutations u to degree 1, which could be done within a few hours of cu time on a standard workstation. Keywords: solution in radicals, roots of unity, cyclotomic olynomials, Galois theory, Gauss algorithm Introduction Niels Henrik Abel and Evariste Galois showed that olynomial equations of degree higher than four cannot be solved by radical exressions in general. As Galois stated in his work, radical solutions exist if and only if the Galois grou of the olynomial is solvable. Since the Galois grou of a cyclotomic olynomial is Abelian, its Galois grou is solvable, and so its solutions can be exressed by radical exressions. Carl Friedrich Gauss roved this secial case already in 1797 and ublished it in 181 in his master-work Disquisitiones Arithmeticae [1] as an extension of his work to determine which regular olygons can be constructed by comass and ruler. In this book Gauss describes an algorithm to comute radical exressions for rimitive roots of unity. We imlemented a variant of the algorithm of Gauss in Male, which has an imroved time comlexity comared to Gauss original roosition. The hard art of the roof and of the algorithm is to comute a radical exression for a rimitive -th root of unity, where is a rime number. The imrovement in our algorithm reduces the asymtotic time comlexity in this case from O(( m 6 )log()) to O( 3 m 6 log()) where m is the largest rime factor of ; 1. An analysis of the algorithm and the statement and roof of the roosition on which the imrovement is based can be found in []. We also refer to this aer for a descrition of the underlying ideas of Gauss algorithm and the statement of the theorems that yield its correctness. Comared to the Male-code given in [], we managed to imrove the ractical seed of the algorithm to a great Arbeitsbereich Symbolisches Rechnen, Fakultät für Informatik, Universität Tübingen, 776 Tübingen, Germany, fweber,keckeiseg@informatik.uni-tuebingen.de; WWW: htt://wwwsr.informatik.uni-tuebingen.de extent, reducing the needed amount of memory at the same time. For instance, the comutations of radical exressions for the 47th, 59th, or 83rd root of unity failed with the revious imlementation, but can now be accomlished within a few hours. The Algorithm First, we will show how the task of comuting radical solutions for cyclotomic olynomials can be reduced to the one of calculating radical exressions for roots of unity. Second, we will sketch the algorithm for the latter task, which mainly is the one given in []. RADICAL SOLUTIONS FOR CYCLOTOMIC POLY- NOMIALS The n-th cyclotomic olynomial over the rational numbers is of the form n (x) = and its degree is (n) = x n ; 1 gcd(x n ; 1 Q n;1 j=1 (xj ; 1)) sy i=1 ( i r i ; i (r i ;1) ) where n = Q s i=1 i r i and is the Eulerian -Function, cf. [3,. 13 and 169]. From the defining formula of it can be seen that the inverse image (;1) of is finite. Thus, to determine whether or not an irreducible factor q of a olynomial is cyclotomic, it suffices to comare the j-th cyclotomic olynomial with q for all j in (;1). Male rovides the function numtheory[invhi] to calculate (;1) ; the comutational costs of this function is negligible for the range of arguments which are feasible for comuting radical exressions. Given the n-th cyclotomic olynomial n (x), the task of comuting (n) radical solutions can be reduced to finding a radical exression for one root of unity, because this root will MaleTech Vol.???, No.??,. 1 5; ISSN $6. c Birkhäuser Boston

2 be rimitive, e. g. a multilicative generator for the others. Further, it suffices to find a rimitive n-th root of unity to comute rimitive roots of unity for all rime factors of n by alying some relatively simle recursion formulas, see e. g. [4]. RADICAL EXPRESSIONS FOR PRIMITIVE -TH ROOTS OF UNITY The hard art of the algorithm is to find a radical exression for a rimitive -th root of unity, where is a rime number. This art involves the comutation of Gaussian eriods, cf. [1, x 343]. If denotes a rimitive -th root of unity the eriod (f k) for two integers f and k is defined by (f k) = f;1 X m= (kg (em) ) where e = ;1 f and g is a rimitive -th root. In Male a rimitive -th root can be comuted by the function rimroot in the ackage numtheory. The Gaussian eriods are generators of intermediate fields of the number field Q( ), which is a number field of degree ; 1 over the rationals. Using the modern terminology of intermediate fields the algorithm of Gauss works roughly as follows. Starting with the rationals, comute a radical exression for a generator of an intermediate field of rime degree, i.e. an aroriate Gaussian eriod. After this task has been accomlished, comute a radical exression for a generator of another intermediate field (i.e. another Gaussian eriod) whose relative degree over the other intermediate field is of rime order. Continue this task until a radical exression for a generator of Q( ), i.e. = (1 g ) has been comuted. The difficult art of Gauss algorithm is the lifting of radical exressions from one intermediate field to another one. We will not give the details of this lifting here but we refer to []. The algorithm described above works for several sequences of intermediate fields to be constructed. All ermutations on the list of rime factors of ; 1 give a ossible sequence in the construction of the intermediate fields. Different ermutations give different results in general, and also the comutational costs are greatly affected by the choice of an aroriate sequence of intermediate fields. Our default strategy in the choice of the intermediate fields is as follows: first take the largest rime factor, then the second largest and so on, until we just get the final field Q( ) as a relative extension of degree of the last intermediate field. Note that we have to count multilicities as searate factors. This strategy seems to be referable over others, because bigger relative extensions which cause more comutations occur in smaller intermediate fields. However, as a tool for exerimental mathematics we rovide another strategy, too. The second strategy is simly to reverse the ordering; this can be done by setting SwaPrimeOrder := true. As had to be exected in all our exeriments the obtained comutation times were far worse than with our default setting and the comuted radical exressions were also larger; some examles are given below in Table. How to Use the Library The library is included in the file radsolvelib. All functionality is available via the function radsolve. The command radsolve serves as an extension to the Male solve command. It returns radical solutions for all cyclotomic factors of a olynomial with integer coefficients and calls solve for the others. Calling Sequence: radsolve(oly) Parameters: oly an univariate olynomial with integer coefficients The command radsolve(oly) returns the solutions of the univariate olynomial oly as radical exressions for all the irreducible factors of oly that are either cyclotomic or of degree lower than four. In general, exlicit solutions in terms of radicals for olynomial equations of degree greater than four do exist if and only if the Galois grou of the olynomial is solvable. This is the case for cyclotomic olynomials, since their Galois grou is cyclic. radsolve(oly) comutes radical exressions for such cases and calls solve for the other factors of oly. In cases, where exlicit solutions can t be comuted, imlicit solutions are given in terms of RootOfs. Note that to obtain exlicit solutions for the general quartic olynomial equation you have to set the global variable EnvExlicit to true, as exlained in the toic hel to solve. The outut from radsolve is a sequence of solutions. The behaving of radsolve(oly) can be changed by using the following global variables (default values in brackets): AllSolutions (false): If AllSolutions is set to false, radsolve(oly) returns only one solution (no matter which exonent belongs to the factor) for every irreducible factor of oly. If it is set to true, all solutions are given. UseRootsymbols (false): if UseRootsymbols is set to true, RootOfs are used to reresent rimitive roots of unity of rime factors of n;1, where n is the degree of MaleTech

3 the cyclotomic olynomial, although these roots could be exressed in radicals exlicitly. This saves time and sace. If UseRootsymbols is set to false, all RootOfs are substituted by their radical exressions. SwaPrimeOrder (false): If SwaPrimeOrder is set to true, then an alternative order of building intermediate generators of field extensions of order ;1 m, where is the degree of the cyclotomic olynomial and m divides ; 1, is chosen. This results in different radical exressions and in more comuting time. Mainly for exerimentation. To obtain information about the comuting rocess, set infolevel[radsolve] to an aroriate level. Level gives just the solutions, level 1 and level are not used, level 3 gives a general outline, level 4 is nice to use for larger examles and level 5 is designed for those who are really interested in the algorithm. EXAMPLES The Male solve command does not exress the solutions of a cyclotomic factor of a olynomial of degree higher than 4 in radicals, but uses sin and cos functions instead. > solve(xˆ7-1)[4]; ;cos( 1 7 ) + I sin( 1 7 ) In the following, we set AllSolutions to false. Thus we get only one solution for any irreducible factor of the olynomial. > lsols:=radsolve(xˆ7-1); lsols := 1 ; 1 (; %1=3 (; (6 ; %1; ;3) (; =3 % ) 1= ; %1= ; %1; ;3 % 1=3 %1 := (; % := 9%1+ 8 ; 6 ;3 ;3) 4 Using the radnormal command we can verify that the radical exression, which was returned by radsolve as a solution for the non-trivial factor, is indeed a 7th root of unity. > radnormal(lsols[]ˆ7-1); To verify the results, one can use radnormal for smaller degrees only. One has to use numeric evaluation for larger degrees. The following method works for very large exressions. > z:=radsolve(numtheory[cyclotomic](41,x)): > Digits:=4: > readlib(otimize): > numeval:= otimize/makeroc ( > ma(evalf,[otimize(z)])): > fnormal(numeval()ˆ41-1,35); Practical Limitations of the Algorithm Comared to [] the imlementation of the main algorithm has been otimized. For results in Table 1 we alied radsolve on all cyclotomic olynomials of (rime) degree u to 11 on a Sun Ultrasarc I workstation. It can be seen that the comuting time deends on the size of the largest rime factor of ; 1 to a great extent, as had to be exected from the theoretical comlexity analysis given in []. Memory usage was u to MB for =83, less than 5MB for the rest. Table 1 was generated semi-automatically via the following scrit. > _UseRootsymbols:=true: > _SwaPrimeOrder:=false: > readlib(cost): > additions:= f : > multilications:= f : > divisions:= f : > functions:= r : > assignments:= a : > readlib(otimize): > for i from 7 to 11 do > if numtheory[isrime](i) then > > t:=round(time( > radsolve( > numtheory[cyclotomic]( > i,x)))): > z:= radsolve(numtheory[cyclotomic](i,x)); > rintf( \\n%5d %15A %f %3A%3A, > i,numtheory[ifactor](i-1), > t,cost(z), > cost(otimize(z))); > fi: > od: It is interesting to see the results you obtain by setting SwaPrimeOrder := true. You get different radical exressions and it takes a lot more time and memory. Some examles are given in Table. Vol.???, No.??,

4 Table 1: Summary of Comutations The following comutations times refer to our Male imlementation of the algorithm on a Sun Ultrasarc I workstation. ; 1 com. size of term size of term time (tree re.) (dag re.) (in sec.) rational radical rational radical oerations oerations oerations oerations Table : Some results with SwaPrimeOrder := true All other settings were equal to the ones used for Table 1. ; 1 com. size of term size of term time (tree re.) (dag re.) (in sec.) rational radical rational radical oerations oerations oerations oerations MaleTech

5 Comarison to Related Work In the book of Gaal [4] a radical exression for a 7th root of unity is derived by some secial reasoning that does not generalize to higher orders. The derived exression is the following: read radsolvelib : > A:=-1/6+(7/+1/*sqrt(-3))ˆ(1/3)/6 > +(7/-1/*sqrt(-3))ˆ(1/3)/6 > +(sqrt((-1/3+(7/+1/*sqrt(-3))ˆ(1/3)/3 > +(7/-1/*sqrt(-3))ˆ(1/3)/3)ˆ-4))/; A := ; ( I 3) 1= ( 7 ; 1 I 3) 1=3 + 1 ( (; ( ; 4) 1= Our function radsolve comutes: I 3) 1= ( 7 ; 1 I 3) 1=3 ) > B:=radsolve(xˆ6+xˆ5+xˆ4+xˆ3+xˆ+x+1); B := ; 1 ; %1=3 (; ;3) (6 ; ;3 ; %1) (; ;3)! 1= ; 1 3 % 1= %1= ; ;3 ; %1 6 % 1=3 %1 := (; % := 9%1+ 8 ; 6 ;3 The radsolve-function comuted the 7th root of unity that is equal to e ( I6 7 ) : > radnormal(a-bˆ6); Although exression A was obtained by a secial method, it is not simler than the one derived by our general algorithm, as the cost function shows: > readlib(cost): readlib(otimize): > cost(otimize(a)); 8 additions + 16 multilications > cost(otimize(b)); + 1 functions + 6 assignments 1 additions + 16 multilications + divisions + 6 functions + 7 assignments Excet for an imlementation by ourselves of a much more inefficient method for the same task described in [5], we do not know of imlementations of other general methods by which radical exressions for higher roots of unity can be comuted. Without being aware of an imlementation, we know of an algorithm develoed by B. Trager, which comutes radical exressions for a -th root of unity [6]. This algorithm is entirely different from the one of Gauss. The major comutational task consists of inverting a matrix of size O() over Q( q ), where q is a divisor of ; 1. Thus if ; 1 is smooth, i.e. if ; 1 contains only small rime factors, the asymtotic time comlexity of our imrovement of the algorithm of Gauss is much better. But in secial cases, such that ;1 is rime, the algorithm of Trager might be an interesting alternative. It would be interesting to have a careful imlementation of the hard cases, such as = etc. Acknowledgment A. Weber was artially suorted by Deutsche Forschungsgemeinschaft under grants We 1945/1-1 and Ku 966/6-1. We are grateful to M. Monagan and an anonymous referee for many detailed comments and suggestions. References [1] Carl Friedrich Gauss. Disquisitiones Arithmeticae. G. Fleischer Jun., Göttingen, 181. In Latin. Rerinted in [7]. German translation: [8]; English translation: [9]. [] Andreas Weber. Comuting radical exressions for roots of unity. SIGSAM Bulletin, 3(117):11, Setember [3] W. Narkiewicz. Elementary and Analytic Theory of Numbers. Sringer-Verlag, 199. [4] Lisl Gaal. Classical Galois Theory. Chelsea Publishing Comany, New York, fourth edition, [5] Harold M. Edwards. Galois Theory, volume 11 of Graduate Texts in Mathematics. Sringer-Verlag, New York, [6] Richard Ziel. Comuter algebra. Unublished Lecture Notes, [7] Carl Friedrich Gauss. Werke, volume I. Georg Olms Verlag, Hildesheim, New York, [8] Carl Friedrich Gauss. Untersuchungen über höhere Arithmetik. Chelsea Publishing Comany, second edition, [9] Carl Friedrich Gauss. Disquisitiones Arithmeticae English Edition. Sringer-Verlag, Vol.???, No.??,