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  1. #1
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    abstract algebra help

    Show that if the field K is generated over F by the elements \alpha_1, \alpha_2, ..., \alpha_n, <br />
, where each \alpha_i is not algebraic over F, then an automorphism \sigma of K fixing F is uniquely determined by \sigma(\alpha_1), \sigma(\alpha_2), ..., \sigma(\alpha_n)<br />
. In particular, show that an automorphism fixes K if and only if it fixes a set of generators for K.

    I think I need to define K, but I'm not sure how to do it. Can anyone help?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Show that if the field K is generated over F by the elements \alpha_1, \alpha_2, ..., \alpha_n, <br />
, where each \alpha_i is not algebraic over F, then an automorphism \sigma of K fixing F is uniquely determined by \sigma(\alpha_1), \sigma(\alpha_2), ..., \sigma(\alpha_n)<br />
. In particular, show that an automorphism fixes K if and only if it fixes a set of generators for K.

    I think I need to define K, but I'm not sure how to do it. Can anyone help?
    are you sure the assumption is not that "the set \{\alpha_1, \alpha_2, \cdots , \alpha_n \} is algebraically independent over F ?" this is very different from "each \alpha_i being transcendental (= not algebraic) over F."
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    are you sure the assumption is not that "the set \{\alpha_1, \alpha_2, \cdots , \alpha_n \} is algebraically independent over F ?" this is very different from "each \alpha_i being transcendental (= not algebraic) over F."
    Yes, it says "do not assume the \alpha's are algebraic over F
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  4. #4
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    Quote Originally Posted by dori1123 View Post

    Yes, it says "do not assume the \alpha's are algebraic over F
    every element of K is in the form u=\frac{P(\alpha_1, \cdots , \alpha_n)}{Q(\alpha_1, \cdots , \alpha_n)}, where P, Q are polynomials in \alpha_1, \cdots , \alpha_n with coefficients in F and Q(\alpha_1, \cdots , \alpha_n) \neq 0. if \alpha_i were algebraic over F, then elements of K

    would have simpler forms, that is in the form P(\alpha_1, \cdots , \alpha_n). anyway, if \sigma is an F automorphism of K, then \sigma(u)=\frac{P(\sigma(\alpha_1), \cdots , \sigma(\alpha_n))}{Q(\sigma(\alpha_1), \cdots , \sigma(\alpha_n))}. now you should be able to solve your problem easily.
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