# automorphisms

• Nov 29th 2009, 10:05 AM
Sampras
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?
• Nov 29th 2009, 05:41 PM
aliceinwonderland
Quote:

Originally Posted by Sampras
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 $\displaystyle \{x^4 -5x^2 + 6\}$.