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Given an ideal $I=\langle f_1,\ldots,f_s\rangle\subseteq\mathbb{C}[X_1,\ldots,X_n]$, suppose that the differentials of its generators $df_1,\ldots,df_s$ are linearly independent at any point $x\in\mathbb{V}_\mathbb{C}(I)$, the algebraic set of $I$ in $\mathbb{C}^n$, can we say that $I$ is a radical ideal?

In general, how do we check if an ideal is radical or not?

Thanks in advance.

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    The answer to the first part is yes. The condition implies the set defined by the $f_i$s is smooth and which in turn implies locally $\mathbb{C}[X_1,\ldots X_n]/I$ is an integral domain. For the second question, there is no general method and the question is too broad. – 2017-02-18
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    There are algorithms for computing the radical of an ideal in a polynomial ring. Here are two sources: [1](https://www.msri.org/~de/papers/pdfs/1992-001.pdf), [2](http://cms.dm.uba.ar/Members/slaplagn/archivos/2006-07_computationOfTheRadical.pdf) – 2017-02-18

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