Suppose . Does the map defined by leave something fixed?

First of all, is the set with the elements and added in? It is an autmomorphism of E?

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- Nov 29th 2009, 11:05 AMSamprasautomorphisms
Suppose . Does the map defined by leave something fixed?

First of all, is the set with the elements and added in? It is an autmomorphism of E? - Nov 29th 2009, 06:41 PMaliceinwonderland
leaves fixed. If F is a field and if and is algebraic over F with , the map defined by

is an isomorphism if and only if and are conjugate over F where . In your example can be thought of as and it is an automorphism in E.

can be thought of as an extension field (actually it is an algebraic extension) of a field adjoining to F the elements and that are not elements of .

can also be thought of as a splitting field of .