i am american
i am an american
i am chinese
are all correct.
If you say:
I am a chinese
it feels like the sentence is incomplete (you are a chinese what?), you would need to say something like:
I am a chinese person.
Typically nationalities that end in '-an' can be used as a noun to mean a person of that nationality (as well as an adjective), e.g. Australian, canadian, american, german etc.
Therefore you can say things like i am an australian, or i am an american.
Other nationalities tend to only act as an adjective, e.g. Chinese, japanese, english, swedish etc.
So sentences like 'i am an english' or 'i am a chinese' sound incomplete because they have no noun.
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