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?



Behaviour of Primes under Regular Coefficient Extensions
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.


