úÎ'!#¥     None A cyclotomic number. The primitive nth root of unity.  For example, (4) = % is the primitive 4th root of unity,  and (5) = exp(2*pi*i/'5) is the primitive 5th root of unity.  In general,  n = exp(2*pi*i/n). The square root of an . The square root of a  number. The square root of -1. Make a Gaussian rational; gaussianRat p q is the same as  p + q * i. 8A complex number in polar form, with rational magnitude r and rational angle s  of the form r * exp(2*pi*i*s);  polarRat r s is the same as  r * e q ^ p,  where s = p/q. Complex conjugate. $Real part of the cyclotomic number. )Imaginary part of the cyclotomic number. 'Step 1 of cyclotomic is gcd reduction. =Step 2 of cyclotomic is reduction to a rational if possible. KCompute phi(n), the number of prime factors, and test if n is square-free. L We do these all together for efficiency, so we only call factorise once. 'Step 3 of cyclotomic is base reduction Sum of two cyclotomic numbers. #Product of two cyclotomic numbers. 6Product of a rational number and a cyclotomic number. Additive identity. Additive inverse. Multiplicative inverse. !Is the cyclotomic a real number? Is the cyclotomic a rational? 'Is the cyclotomic a Gaussian rational? %Export as an inexact complex number. .Export as an inexact real number if possible. -Return an exact rational number if possible. (Sine function with argument in degrees. *Cosine function with argument in degrees. The golden ratio, (1 + "5)/2. Discrete Fourier transform,  X_k = sum_{n=0}^{N-1} x_n c dot e^{-i 2 pi f rac{k}{N} n}. $Inverse discrete Fourier transform,  x_n = f rac{1}{N} sum_{k=0}^{N-1} X_k c dot e^{i 2 pi f rac{k}{N} n}. !abs and signum are partial functions. N A cyclotomic number is not guaranteed to have a cyclotomic absolute value.  When defined, signum cI is the complex number with magnitude 1 that has the same argument as c;   signum c = c / abs c. B"#$%&'()*+, -./0123456789:;<=>? @ABC!  ?"#$%&'()*+, -./0123456789:;<=>? @ABC!D      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIcyclotomic-0.4Data.Complex.Cyclotomic Cyclotomice sqrtIntegersqrtRati gaussianRatpolarRatconjrealimagisRealisRat isGaussianRat toComplextoRealtoRatsinDegcosDeg goldenRatiodftdftInv integer-gmpGHC.Integer.TypeIntegerbaseGHC.RealRational gcdReduce tryRational phiNrpSqfree tryReducesumCycprodCyc prodRatCyczeroCycaInvCycinvCyc$fNumCyclotomicordercoeffs showBaseExp leadingTermfollowingTerms followingTermshowRatebsqrt2sqrtPositiveIntegerabsValsigNum convertToBase removeZeros cyclotomic mkCyclotomicgcdCycequalCoefficientslenCyc reduceByPrimeequalReplacements replacementsreplace includeMods removeExpspqPairsextraneousPowersmultiplyExponentsproductOfGaloisConjugatesgcdListequal$fShowCyclotomic$fFractionalCyclotomic