Cyclotomic polynomials

There is a simple device to detect repeated occurrence of a factor in a polynomial with coefficients in a field.

Let k be a field. For a polynomial

f(x) = cnxn + . . .+ c1x+ c0

with coefficients ci in k, define the (algebraic) derivative [1] Df(x) of f(x) by

Df(x) = ncnxn−1 + (n− 1)cn−1xn−2 + . . .+ 3c3x2 + 2c2x+ c1 Better said, D is by definition a k-linear map

defined on the k-basis {xn} by D(xn) = nxn−1

8.1.1 Lemma: For f, g in k[x],

D(fg) = Df · g + f ·Dg

8.1.2 Remark: Any k-linear map T of a k-algebra R to itself, with the property that

T (rs) = T (r) · s+ r · T (s)

is a k-linear derivation on R.