How to prove subfield generated $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n) $

Question

This is problem in Hungerford chapter 5: Fields and Galois Theory.

Prove $K(u_1,u_2,..u_{n-1},u_n)=K(u_1,u_2,..u_{n-1})(u_n)$ and $K[u_1,u_2,..u_{n-1},u_n]=K[u_1,u_2,..u_{n-1}][u_n] $

My teacher skipped chapter 4 of the book: Modules and I think this is the reason I am unable to do it.

Answer 1

Just use the definition: $K(u_1,\dots ,u_n)$ is an extension field of $K(u_1,\dots ,u_{n-1})$ which contains $u_n$, so $K(u_1,\dots ,u_{n-1})(u_n) \subset K(u_1,\dots ,u_n)$. On the other hand, $K(u_1,\dots ,u_{n-1})(u_n)$ is an extension of $K$ containing $u_1, \dots , u_n$, so $K(u_1,\dots ,u_n) \subset K(u_1,\dots ,u_{n-1})(u_n)$.

Obviously, switching $[$ $]$ for $($ $)$ and "ring" for "field" will work for the second part.