What is the Degree of Field Extensions in Quotient Fields?

[cap.r]

Homework Statement

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?

Homework 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

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).

 

[ystael]

Have you tried the direct approach?

Suppose without loss of generality that \deg f < \deg g. There is a fairly simple set of \deg g elements of K(t) which you might try to prove is a basis for K(t) over K(u).

 

[cap.r]

I am not sure if i understand the question anymore after reading your hint. what do elements in K(u) look like?