Question

Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring expand to a prime ideal in the second?

Answer 1

The answer is Yes.

A field extension $K \to K'$ is regular if and only if for every $K$-algebra $A$ which is an integral domain the $K'$-algebra $A' := A \times_K K'$ is an integral domain (Bourbaki, Algebra, Chap. V, §17, No.3). Applying this to $A = K[x_1,\dots,x_n]/{\mathfrak p}$ for a prime ideal ${\mathfrak p}$ we obtain $A' = K'[x_1,\dots,x_n]/{\mathfrak p}K'[x_1,\dots,x_n]$. Thus ${\mathfrak p}K'[x_1,\dots,x_n]$ is a prime ideal.