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Whenever you post provide your full sql developer version, full DB version and OS version and platform.
how can I create a local connection and local Oracle 11g database using SQL developer on windows vista laptop?
First, if you haven't done so, you need to download and install Oracle 11g. When you install it you will be ask if you want to create a database.
The easiest way to create a database after installation is to use DBCA (database configuration assisantant).
After you install Oracle use sql*plus to confirm that the DB is running and that you can connect to it.
Then download and install the latest version of sql developer.
This Oracle article covers how to create connections in sql developer