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There is no european nationality. You have your own nationality (french, italian, etc), and based on this nationality you can traval or work or live in any of european countries. There are some agreements with some countries, not member of the eu area, such as Switzerland.
If yoy are French already, there is no problem for yoy to enter any of these countries.
This reply is much simplified, but it is applicable in most of european countries.
I hope this helps