## Problems in Section 1 of GTM 167

Please post carefully solutions of the following ones:

1. Let $K$ be a field extension of $F$. By defining scalar multiplication for $\alpha\in F$ and $a\in K$ by $\alpha\cdot a=\alpha a$, the multiplication in $K$, show that $K$ is an $F-$ vector space.

2. If $K$ is a field extention of $F$, prove that $[K:F]=1$ iff $K=F$.