A good starting point for research into your immigrant ancestor's citizenship is the U.S. Federal Census, which asked questions of naturalization beginning in 1820, although individuals other than the head of household were not named until 1850; in some cases the questions applied only to white males of a certain age, so it's important to notice what question is being asked, and of whom. The 1920 census was the first and last year to ask specifically for the Year of Naturalization: it also asks the Year of Immigration, which can help in calculating waiting periods, etc.
Tracing your immigrant ancestor through every census year from the first census after birth to the last census before death, is a good way to determine naturalization and citizenship. And it's important to distinguish between the two: not everyone who became a U.S. citizen was naturalized. Wives and children often derived citizenship from their husband or father, so if he was a U.S. citizen by birth or was naturalized, his wife and children became citizens by default.
If your immigrant ancestor was naturalized, you can search for naturalization papers, although it may take a bit of sleuthing. If not naturalized, it's possible your ancestor was subject to alien registration under one of the Alien Acts or Enemy Alien Registrations required at various times in the U.S. as a national security measure.
Keep in mind, naturalization could also denied by race and circumstance, leaving out indentured servants, slaves, free blacks and Asians. The American Indian was permitted citizenship on a selective basis until 1924. And even single women could not apply whose father had never been resident in the United States. So the rules were complex for different groups at different times, but the census may be of help in sorting it out.
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