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Multiple root of a polynomial

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October 31, 2010

algebrapolynomialalgebra unsolved

Problem Statement

Let

\delta

be a symbol such that

\delta \neq 0

and

\delta^2 = 0

. Define

\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}

, where

a+ b \delta = c+ d \delta

if and only if

a = c

and

b = d

, and define

(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,

(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.

Let

P(x)

be a polynomial with real coefficients. Show that

P(x)

has a multiple real root if and only if

P(x)

has a non-real root in

\mathbb R[\delta].

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