Cyclotomic

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

For fixed z

Values at infinities

General characteristics

Domain and analyticity

(Subscript[C, n])(z) is a polynomial of z and as such an analytical function of z. (Subscript[C, n])(z) is defined in the whole complex z-plane and for integer n.

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to z

The function (Subscript[C, n])(z) has a pole of order φ(n) at z==(OverTilde[∞]).

Branch points

With respect to z

The function (Subscript[C, n])(z) does not have branch points.

Branch cuts

With respect to z

The function (Subscript[C, n])(z) does not have branch cuts.

Series representations

Generalized power series

Expansions at z==0

Product representations

Generating functions

Transformations

Multiple arguments

Power of arguments

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Involving only one direct function

Products

Finite products

Inequalities

Zeros

Theorems

Degree of (Subscript[C, n])(x)

(Subscript[C, n])(x) are irreducible polynomials over ℤ with degree φ(n) .

Coefficients of (Subscript[C, n])(x)

(Subscript[C, 105])(x) has coefficients of -2 for (x^7) and (x^41), making it the first cyclotomic polynomial to have a coefficient other than ±1 and 0 (because 105 is the first number to have three distinct odd prime factors, i.e., 105==3×5×7). The coefficients of (Subscript[C, p q])(x) for distinct primes p and q can be only  0, ±1. The smallest values of n for which (Subscript[C, n])(x) has one or more coefficients  ±2,±3,… are 105,  385,  1365,  1785,  2805,  3135,  6545, ….

History

– L. Euler (1771) studied cases (Subscript[C, n])(x)==0,n<10 
– A.-T. Vandermonde (1771) studed (Subscript[C, 11])(x)==0
– J.-L. Lagrange (1772)
– C. F. Gauss (1796) studed (Subscript[C, 17])(x)==0
– N. H. Abel (c. 1824, published in 1829) showed that (Subscript[C, n])(x)==0 can always be solved in radicals
– L. Kronecker (1845)
– F.G.M. Eisenstein (1850)
– A. Migott (1883)
– P. Bachmann (1872)
– L. Carlitz (1936)
– E. Lehmer (1936)

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