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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.
     
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