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Math Help - extension

  1. #1
    Senior Member Sampras's Avatar
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    extension

    Let  E be an algebraic extension of a field  F , and let  \sigma be an automorphism of  E leaving  F fixed. Let  \alpha \in E . Show that  \sigma induces a permutation of the set of zeros of  \text{irr}(\alpha, F) that are in  E .

    Let  \text{irr}(\alpha, F) = a_0+a_{1}x+ \cdots + a_{n}x^n . Then  a_0+ a_{1}\alpha + \cdots + a_{n} \alpha^{n} = 0 . So  a_0+ a_1 \sigma(\alpha) + \cdots + a_{n} \sigma(\alpha)^{n} = \sigma(0) = 0 . So  \sigma(\alpha) is a zero of  \text{irr}(\alpha, F) .
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Let  E be an algebraic extension of a field  F , and let  \sigma be an automorphism of  E leaving  F fixed. Let  \alpha \in E . Show that  \sigma induces a permutation of the set of zeros of  \text{irr}(\alpha, F) that are in  E .

    Let  \text{irr}(\alpha, F) = a_0+a_{1}x+ \cdots + a_{n}x^n . Then  a_0+ a_{1}\alpha + \cdots + a_{n} \alpha^{n} = 0 . So  a_0+ a_1 \sigma(\alpha) + \cdots + a_{n} \sigma(\alpha)^{n} = \sigma(0) = 0 . So  \sigma(\alpha) is a zero of  \text{irr}(\alpha, F) .
    correct. you should probably explain it a little bit more if it's a part of your homework.
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