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Math Help - Galois groups

  1. #1
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    Galois groups

    Let G \leq Gal(K/F) be a subgroup of the Galois group of the extension K/F and suppose \sigma_1, \sigma_2, ..., \sigma_n are generators for G. Show that the subfield E/F is fixed by G if and only if it is fixed by the generators \sigma_1, \sigma_2, ..., \sigma_n.

    I think the forward direction is clear. Since G fixes E and since the generators \sigma_1, \sigma_2, ..., \sigma_n are in G, then these generators must fix E.
    I am not sure how to prove the backward direction. Some help please.
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    Let G \leq Gal(K/F) be a subgroup of the Galois group of the extension K/F and suppose \sigma_1, \sigma_2, ..., \sigma_n are generators for G. Show that the subfield E/F is fixed by G if and only if it is fixed by the generators \sigma_1, \sigma_2, ..., \sigma_n.

    I think the forward direction is clear. Since G fixes E and since the generators \sigma_1, \sigma_2, ..., \sigma_n are in G, then these generators must fix E.
    I am not sure how to prove the backward direction. Some help please.
    i think the reason that you can't solve an easy problem like this is that you don't know what is meant by "G is generated by \{\sigma_1, \cdots , \sigma_n \}. well, here is the meaning:

    every element of G is in the form x_1^{r_1}x_2^{r_2} \cdots x_m^{r_m}, where m is any natural number, r_i \in \mathbb{Z}, and x_i \in \{\sigma_1, \cdots , \sigma_n \}.
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