Proposition 7

Tag 06B0, permanent. Cite it as [Bourbaki, Tag 06B0].

Algebra, chapter IV, § 2. Zeros of polynomials

— Let $f \in \mathbf{A}[X]$ and let $a \in \mathbf{A}$ be a root off: For $a$ to be a simple root off it is necessary and sufficient that $a$ should not be a root of the derivative $Df$ of $f$ .

By hypothesis we have $f = (X - a)\ g$ , where $g \in \mathbf{A}[X]$ . For $a$ to be a simple root of $f$ it is necessary and sufficient that $g(\alpha) \neq 0$ . Now we have $Df = g + (X - \alpha)\ Dg$ , whence $(Df)(\alpha) = g(\alpha)$ .

More generally:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Proposition 3, § 12
Proposition 4, § 16
Proposition 4, § 7

Cites

Nothing.