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Uniqueness of Splitting Fields

  1. Oct 23, 2004 #1
    In my class notes, I have two theorems which don't quite seem to fit together. Maybe you can help me out.
    Thm 1 If p(x) in F[x] splits in K, then E=F(a1,...,an) is the splitting field of p(x) in K (the a_i's are the roots of p(x)).
    Thm 2 If p(x) in F[x], then the splitting field of p(x) is unique up to isomorphism.
    I'm clearly missing something big here. Doesn't (1) imply (2)? Isn't (1) even stronger than (2)?
    What's an example of a polynomial with two distinct but isomorphic splitting fields?
     
  2. jcsd
  3. Oct 24, 2004 #2

    matt grime

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    Strictly speaking Thm 1 should state that E is *a* splitting field for p.

    Thm 2 then states that this is essentially unique.

    For instance, it is not obvious, but it is true, that

    [tex] \mathbb{Q}[\sqrt{3},\sqrt{5}] \cong \mathbb{Q}[\sqrt{3}+\sqrt{5}][/tex]

    So that [tex]\mathbb{Q}[\sqrt{3}+\sqrt{5}][/tex] is the splitting field of

    [tex](x^2-3)(x^2-5)[/tex] and isn't of the form you wrote.
     
  4. Oct 24, 2004 #3

    mathwonk

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    Matt's point is that although you are right, the definition of the splitting field in K, makes it opbvious that there is only one such field IN K, there may be other splitting fields that are not in K.

    Matt's example is a little misleading to me since it is the same field but just written with a different generator. It would persuade me more if he were to give an isomorphic splitting field not lying in the same ambient field, such as Q[X] modded out by the minimal polynomial of sqrt(3)+sqrt(5).
     
  5. Oct 24, 2004 #4

    matt grime

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    Feel free to post a better example.
     
  6. Oct 24, 2004 #5
    Ok... This is helping. I guess the problem is this:
    Suppose we have a field F and two extension fields K and K' in which a polynomial p(x) splits. We have the splitting field E for p(x) in K and the splitting field E' for p(x) in K'.
    The problem in my mind is that I'm tempted to say E=E'. I mean, when we mod out by a certain irreducible, we always view F as contained in the resulting field. But if K and K' both are extension fields, and we view them as containing F, then E and E' both contain F. E and E' are both supposedly the smallest subfields of K and K' containing the roots of p(x). But their intersection is a field contained in K and K', and containing the roots of p(x). So the intersection is in fact equal to E and E' and so E=E'.
    This is why I'm troubled. I guess it's just a silly point. I know there are instances when splitting fields aren't actually equal, but it certainly seems obvious to me that they should be isomorphic (in your example above, it wasn't as obvious, but I think that's partly because it wasn't even obvious that that polynomial even splits in Q[root3+root5]). But the proof given in class was much more complex, and seemingly unnecessarily so.
     
    Last edited: Oct 24, 2004
  7. Oct 25, 2004 #6

    matt grime

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    The proof I know for the isomorphism of splitting fields is trivial, and roughly says what you just said, but in fewer words. Although that is probably a function of the order and style in which I learned the results. However, there is also the stronger result that splitting fields behave well with respect to field isomorphisms.
     
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