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Math Help - automorphisms

  1. #1
    Senior Member Sampras's Avatar
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    automorphisms

    Suppose  E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) . Does the map defined by  \sigma(a+b \sqrt{2} + c \sqrt{3}+d \sqrt{6}) = a+ b\sqrt{2}-c \sqrt{3}- d \sqrt{6} leave something fixed?

    First of all,  E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) is the set  \mathbb{Q} with the elements  \sqrt{2} and  \sqrt{3} added in? It is an autmomorphism of E?
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  2. #2
    Senior Member
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    Quote Originally Posted by Sampras View Post
    Suppose  E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) . Does the map defined by  \sigma(a+b \sqrt{2} + c \sqrt{3}+d \sqrt{6}) = a+ b\sqrt{2}-c \sqrt{3}- d \sqrt{6} leave something fixed?

    First of all,  E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) is the set  \mathbb{Q} with the elements  \sqrt{2} and  \sqrt{3} added in? It is an autmomorphism of E?
    \sigma leaves \mathbb{Q}(\sqrt{2}) fixed. If F is a field and if \alpha and \beta is algebraic over F with deg(\alpha, \beta)=n, the map \psi_{\alpha, \beta}:F(\alpha) \rightarrow F(\beta) defined by

    \psi_{\alpha, \beta}(c_0 + c_1\alpha+ \cdots +c_{n-1}\alpha^{n-1})= c_0 +c_1\beta + \cdots +c_{n-1}\beta^{n-1}

    is an isomorphism if and only if \alpha and \beta are conjugate over F where c_i \in F. In your example \sigma can be thought of as \psi_{\sqrt{3}, -\sqrt{3}} and it is an automorphism in E.

     E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) can be thought of as an extension field (actually it is an algebraic extension) of a field \mathbb{Q} adjoining to F the elements \sqrt{2} and \sqrt{3} that are not elements of \mathbb{Q}.
     E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) can also be thought of as a splitting field of \{x^4 -5x^2 + 6\}.
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