# The same result for $\mathbb{C}$ is true for algebraically closed field?

The following result about polynomials is known:

Proposition: Let $$K$$ be a subfield of $$\mathbb{C}$$, $$f(x) \in K[x]$$ a polynomial with degree $$n \geq 1$$ and $$\alpha \in \mathbb{C}$$ a root of $$f(x)$$. Then

a) $$\alpha$$ is a simple root of $$f(x)$$ $$\iff$$ $$f(\alpha) = 0$$ and $$f'(\alpha) = 0$$;

b) if $$f(x)$$ is irredutible over $$K$$ then all the roots of $$f(x)$$ are simple.

Is this result true if we exchange $$\mathbb{C}$$ for any algebraically closed field?

I believe you want to say that $$\alpha$$ is a simple root implies that $$f(\alpha)=0, f'(\alpha)\neq 0$$. This is not true is the characteristic is $$p$$, $$X^p-a=(X-a)^p$$ and $$a$$ is simple.
• It is a subfield of $\mathbb{C}$ Char=0 Commented Jul 1, 2020 at 23:54
• an algebraically closed field is not always a subfield of $\mathbb{C}$. Commented Jul 1, 2020 at 23:55
• I thought you missed read that $K$ is subfield of $\mathbb{C}$ but it seems I missed read the questions Commented Jul 2, 2020 at 0:17