MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2) explains in an intuitive way why if $L$ is a finite separable extension of $K$, the map $(x,y) \mapsto Tr(xy)$, where $x,y$ are in $L$, is a non degenerated bilinear form on L?