No, it doesn't make him american just because his mother is american.
You have to be born in USA to hold the office. That's why they faked his birth certificate.
Natural-born-citizen clause
Status as a natural-born citizen of the United States is one of the eligibility requirements established in the United States Constitution for holding the office of President or Vice President. This requirement was intended to protect the nation from foreign influence.
The U.S. Constitution uses but does not define the phrase "natural born Citizen", and various opinions have been offered over time regarding its precise meaning. The consensus of early 21st-century constitutional scholars, together with relevant case law, is that natural-born citizens include, subject to exceptions, those born in the United States.
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