Resident aliens, in general, have the same liability for Social Security/Medicare Taxes that U.S. Citizens have. Therefore, if working in the U.S. amounts for Social Security should be deducted from your paycheck. If you work for yourself there is no paycheck from which Social Secuity taxes can be deducted - perhaps that is your situation. The Internal Revenue Code provides several exemptions for foreign nationals from social security taxes. If an exemption is unavailable the foreign national...
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Perhaps it would be "OK" to keep your name the way it is, but if you work under a different name than the one on your Social Security card, there could be problems later when you need to collect benefits from the Social Secuirity Administration. Best thing to do is go on line with Social Security (www.ssa.gov) and change your name with them to your "legal last name."