# Degree of field extensions

1. Mar 4, 2010

### cap.r

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).

2. Mar 4, 2010

### 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 $$K(u)$$.

3. Mar 6, 2010

### cap.r

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