# principal ideal notation [closed]

Let $I = \{f(X) = \mathbb{C}[X] | f(0) = f(1) = f(−1) = 0\}$. Then $I$ is an ideal of $\mathbb{C}[X]$.

Deduce that $I = (X^3 −X)$ is the principal ideal generated by $X^3 −X$.

Can someone write $(X^3 −X)$ in set notation because I have no idea what it actually is.

• $(X^3-X)$ denotes the set of multiples of $X^3-X$ by another polynomial. It is an ideal because it is stable by sums and multiplication by elements of $\mathbf C[X]$. Explicitly: $\,(X^3-X)=\bigl\{P(X)(X^3-X)\mid P(X)\in \mathbf C[X]\bigr\}$. Apr 13 '15 at 18:09
• I meant like if we have, $S = (m)= m\mathbb{Z} = \{mx | x ∈\mathbb{Z}\}$. What would it be in this case? Apr 13 '15 at 18:11

$$\langle X^3 - X\rangle = \{p(X)\dot\, (X^3 - X); p(X) \in \mathbb K[X]\}$$ is the ideal generated by $X^3 - X \in \mathbb K[X]$.
• Any field $\mathbb K$. You may take $\mathbb K = \mathbb C$. Apr 13 '15 at 18:10