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Math Help - Algebraic number field basis

  1. #1
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    Algebraic number field basis

    Hi! I tried to show the following but i failed: We have a finite galois extension L/\mathbb{Q} with commutative galois group and O_L the ring of algebraic integers in L. Suppose there exists x \in O_L so that the set \{\sigma(x): \sigma \in Gal( L/\mathbb{Q})\} forms a \mathbb{Z}-Basis of O_L.
    Then for every field K \subset L there also exists y \in O_K so that \{\phi(y): \phi \in Gal( K/\mathbb{Q})\} forms a \mathbb{Z}-Basis of O_K.

    I could show that K/\mathbb{Q} is normal and hence each \phi \in Gal (K/\mathbb{Q}) is just a restriction of \sigma \in Gal (L/\mathbb{Q}). But i dont know how to continue the argument. Can anybody please help me?

    Greetings
    Banach
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  2. #2
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    Quote Originally Posted by Banach View Post
    forms a \mathbb{Z}-Basis of O_L.
    I want to help but I do not what a Z-basis is.
    Can you tell me?
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  3. #3
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    Hi! It is very nice that you want to help me. Here it just means that every a \in O_L can be written as a=\sum \limits_{i=1}^n \lambda_i \sigma_i(x) with \lambda_i \in \mathbb{Z} and \sigma_i the elements of the galois group.
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  4. #4
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    Another idea: It also follows that K is galois, so maybe the primitive element theorem helps?
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