1. The problem statement, all variables and given/known data so this is a challenge problem that I need help getting started with. given K a field and K(t) a quotient field over K. let u=f/g for f,g in K(t). IF [K(t):K(u)] is finite then it is equal to max(deg f, deg g). Why is this true? 2. Relevant equations K(t)----K(u)----K [K(t):K] is infinite obviously since t is transcendental over K. I can use anything up to Galois theory and although we didn't cover splitting fields yet, I don't think he will mind if i use them as long as it helps 3. The attempt at a solution as an example I came up with this. f=t^2+1, g=t^3+t+1. then it is easy to see that [K(t):K(u)]=3 which is the max(deg f, deg g).