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?
[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
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).