1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sanity Check for Simple Extension Proof

  1. Dec 1, 2013 #1
    This was an exercise out of Garling's A Course in Galois Theory.

    Suppose ##L:K## is a field extension. If ##[L:K]## is prime, then ##L:K## is simple.

    I've developed a habit of checking my work for these exercises religiously (the subject matter is gorgeously elegant, so I want to do it justice). I found a much more complicated answer than mine when I googled it, which makes me uncomfortable.

    Here's my proof:

    Suppose ##\alpha\in L## is not in ##K##. Because ##[L:K]=[L:K(\alpha)][K(\alpha):K]## and ##[K(\alpha):K]\neq 1##, we must have ##[K(\alpha):K]=[L:K]##. Because any two vector spaces with the same dimension over the same field are isomorphic, this completes the proof. []

    Is this right?
     
  2. jcsd
  3. Dec 1, 2013 #2
    This should be in the homework forum by the way.

    And yes, the proofs seems correct to me.
     
  4. Dec 1, 2013 #3
    Oops! Indeed, it should. My apologies.

    Thank you.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Loading...