Degree of field extensions

  • Thread starter cap.r
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  • #1
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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).
 

Answers and Replies

  • #2
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Have you tried the direct approach?

Suppose without loss of generality that [tex]\deg f < \deg g[/tex]. There is a fairly simple set of [tex]\deg g[/tex] elements of [tex]K(t)[/tex] which you might try to prove is a basis for [tex]K(t)[/tex] over [tex]K(u)[/tex].
 
  • #3
67
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I am not sure if i understand the question anymore after reading your hint. what do elements in [tex]K(u)[/tex] look like?
 

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