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Thread: automorphisms

  1. #1
    Senior Member Sampras's Avatar
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    automorphisms

    Suppose $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $. Does the map defined by $\displaystyle \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, $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $ is the set $\displaystyle \mathbb{Q} $ with the elements $\displaystyle \sqrt{2} $ and $\displaystyle \sqrt{3} $ added in? It is an autmomorphism of E?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Suppose $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $. Does the map defined by $\displaystyle \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, $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}) $ is the set $\displaystyle \mathbb{Q} $ with the elements $\displaystyle \sqrt{2} $ and $\displaystyle \sqrt{3} $ added in? It is an autmomorphism of E?
    $\displaystyle \sigma$ leaves $\displaystyle \mathbb{Q}(\sqrt{2})$ fixed. If F is a field and if $\displaystyle \alpha$ and $\displaystyle \beta$ is algebraic over F with $\displaystyle deg(\alpha, \beta)=n$, the map $\displaystyle \psi_{\alpha, \beta}:F(\alpha) \rightarrow F(\beta)$ defined by

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

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

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