On a theorem of Nagata’s UFD criterion

Question

In this paper Nagata proves that if char$K$ ≠ 2 then $K[X_1,...,X_n]/(\sum X_i^2)$ is UFD for $n \geq 5$. There I have a doubt in the proof of the theorem 2. He is taking a height 1 prime ideal $p$ in $K[X_1,..,X_n]/(\mathfrak{a})$. Then why the ideal $pL[X_1,...,X_n]/(\mathfrak{a})$ has no embedded prime ? If $L$ is algebraic over $K$ then I can see it is true because of integral extension.

Thanks..