Cyclotomic polynomial for n=0

[franktaw at netscape.net]

Strictly speaking, the cyclotomic polynomial for n=0 is undefned.  The 
two packages have chosen different ways to extend the concept to n=0; 
it would have been perfectly OK for them to have returned some kind of 
error.

There are reasonable arguments for either choice.  A lot comes down to 
the basic property that the degree is phi(n); the question then is, 
what is phi(0)?  Again, this is strictly speaking undefined, but there 
are two reasonable choices: 0 and 1.  phi(0) = 0 gives 1 as the value 
of the 0th cyclotomic polynomial.  If we take phi(0) = 1, then we need 
to choose a root for the polynomial.  Choosing 0 as the root gives the 
nice property that the cyclotomic polynomials for n >= 0 constitute all 
the primitive monic polynomials all of whose roots lie in the unit 
disk; hence the selection of x as the polynomial.

If you take the formula Phi_n(x) = Product_{0<=k<n, gcd(k,n) = 1} e^{2 
k Pi i / n}, you get Phi_0(x) = 1.  On the other hand, if we regard 
this as a product over distinct residues modulo n that are relatively 
prime to n, we get x - e^{2 Pi i / 0}, which is undefined.

You can also make a case for 0 as the "correct" value.  Phi_n(x) is 
x^n-1, divided by all x - u for primitive dth roots of unity u for d a 
proper divisor of n.  x^0-1 is 0; you can divide it by anything you 
want, and the result is still 0.  (Of course, this becomes an infinite 
product in the denominator - but this is only a plausibility argument.)

Franklin T. Adams-Watters


-----Original Message-----
From: noe at sspectra.com

For n=1,2,3,... the cyclotomic polynomials are

x-1, x+1, x^2+x+1,...

It seems that there are two conventions for n=0:

x or 1

Mathematica says it is 1.  I think Maple uses x.  Can someone explain 
this?

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