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Math Help - field extension, gcd

  1. #1
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    field extension, gcd

    Suppose K/F is a field extension of degree m and that \alpha \in K. Prove that for any integer n such that \text{gcd}(m, n)=1, F(\alpha)=F(\alpha^n).

    I was thinking initially in this problem to use the tower lemma but nothing seemed to work out after that. I do not know how to show that for any integer n such that \text{gcd}(m, n)=1, F(\alpha)=F(\alpha^n). Thanks in advance.
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  2. #2
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    Quote Originally Posted by eskimo343 View Post
    Suppose K/F is a field extension of degree m and that \alpha \in K. Prove that for any integer n such that \text{gcd}(m, n)=1, F(\alpha)=F(\alpha^n).

    I was thinking initially in this problem to use the tower lemma but nothing seemed to work out after that. I do not know how to show that for any integer n such that \text{gcd}(m, n)=1, F(\alpha)=F(\alpha^n). Thanks in advance.
    This is the sketch of the proof. I'll assume K=F(\alpha) where [F(\alpha):F]=m. Since the degree of extension is finite, it is an algebraic extension. Then, 1, \alpha, \alpha^2, \cdots, \alpha^m is a basis for K as a vector space over F. If gcd(m,n)=1, then you see that F(\alpha^n) has the same basis set with 1, \alpha, \alpha^2, \cdots, \alpha^m because the field generated by \alpha over F and the field generated by \alpha^n over F is the same.
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