How to hide file extensions
Solution 1:
As far as I know there is no way to acheive this. The Nautilus file manager doesn't seem to have this option in its preferences and for good reason.
File extensions are fake
By which I mean, in Linux, they are simply part of the file name and have no special meaning like in Windows. In Windows, there is a distinction between file names and their extensions whereas in Linux, the 'extension' is just the part of the name, if any, after a '.'. Linux allows the '.' character to be used as part of the file name but Windows reserves it for separating the file extension. Windows uses this file extension to determine how the file should behave but Linux uses MIME types.
This means that there is no reliable way for a file manager to hide the file extension - '.' can be used anywhere in a file name so hiding file extensions may hide part of the file name - this behaviour would be confusing. File extensions are only used to make it easier for a user to tell what type of file they are dealing with - hiding this would hide useful information.
Solution 2:
Other than in Windows file extensions are not needed for files in Ubuntu. However they are used for convenience when it comes to sort out files according to their content. Therefore extensions are more like a fixed part of the filename and thus will always be displayed.