Create SQL table allow NULL(unique key?)

This is my code:

CREATE TABLE table(
        id int(1) UNIQUE,
        name varchar(128) NOT NULL)

 ALTER TABLE table ADD UNIQUE KEY id (id);

I'm not sure if I need the second statement or not, but either way it doesn't work when inserting INSERT INTO table( name ) VALUES( "test" ). It says table.id may not be NULL. What do I need to do to make it work with NULL(and Auto Increment?) Replacing the first column with id int(1) NOT NULL AUTO_INCREMENT, makes it say syntax error.