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As I have been going through Ancestry to work on my family tree, I have come across a search for specifically what you are asking. Yes, the children would automatically gain citizenship once their parents naturalized. If you know his name you can always go to the Ellis Island page and look up what ship he came in on and that may help you get more details that you don't already have. I hope this has helped you.
Beth, thank you very much for the reply. I will check into the sources you suggested. joan
Dear Ms. Stachnik,
Thank you for contacting History Hub!
FamilySearch has a wonderful web page explaining the naturalization and citizenship process, as well as, describing what records are available for what locations and time periods.
There is a section that discusses children. It says that children receive derivative citizenship from their parents. It also states that no extra paperwork was created and that in order to prove citizenship, they would use their parent’s certificate of naturalization. It goes on to say that starting in 1929, individuals that derived their citizenship through their parents could apply for and receive a Certificate of Citizenship in their own name. There are instructions on how to search for and obtain a copy of a Certificate of Citizenship for an individual through U.S. Citizenship and Immigration Services.
Yes, infants should be listed on a passenger manifest.
Best of luck in your research!