Lemma 35.19.1. If $f : X \to Y$ is a flat and surjective morphism of schemes and $X$ is reduced, then $Y$ is reduced.

Proof. The result follows by looking at local rings (Schemes, Definition 26.12.1) and Algebra, Lemma 10.164.2. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).