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Basis for an extension field

  1. Mar 31, 2010 #1
    I am having no luck understanding how to find the basis of a field adjoined with an element.

    For example
    I know that if i take a=sqrt(2)+sqrt(3) that i can find a polynomial (1/4)x^4 - (5/2)x^2 + 1/4 that when evaluated at a is equal to zero.

    So, from that I know the degree is 4 and the basis should have 4 dimensions.

    {1, sqrt(2), sqrt(3), Y} where Y is the part I don't understand.
    {1, sqrt(2), sqrt(3), sqrt(6)} is the actual basis, but how do you get sqrt(6) as Y?

    I don't see how it is a linear combination of sqrt(2) and sqrt(3) as defined in the book I'm using. Since it should be a additive combination of scalars from Q times sqrt(2) and sqrt(3)... but the only way to obtain sqrt(6) is sqrt(2)*sqrt(3), in which neither are scalars as elements of Q.

    Any help understanding this concept is greatly appreciated.
  2. jcsd
  3. Mar 31, 2010 #2
    Remember, you should not be able to get sqrt(6) as a Q-linear combination of sqrt(2) and sqrt(3) for otherwise the set {1,sqrt(2), sqrt(3), sqrt(6)} would be dependent and therefore not a basis. You know that sqrt(6) is in your field however (being the product of sqrt(2) and sqrt(3)) so if you can show it is independent from sqrt(2) and sqrt(3) over Q you will have your basis (since you already know the dimension is 4).
  4. Mar 31, 2010 #3
    So by putting sqrt(2) and sqrt(3) adjoined to the field, then it has to have multiplicative closure so sqrt(6) has to be there... so simple now that I think about it.

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