16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 4.5.3

Polynomials

Mathematics — Linear Algebra · distinct-roots-relatively-prime-derivative

Problem

Let

D

be the differentiation operator on the space of polynomials over the field of complex numbers. Let

f

be a monic polynomial over the field of complex numbers. Prove that

f = (x - c_1) \cdots (x - c_k)

where

c_1, \ldots, c_k

are distinct complex numbers if and only if

f

and

Df

are relatively prime. In other words,

f

has no repeated root if and only if

f

and

Df

have no common root. (Assume the Fundamental Theorem of Algebra.)