There are several references in the Torah which allude to G-d's prohibition of interfaith marriages. In Deuteronomy 7:1-5, it says "...(the non-Jewish male marriage partner) will cause your child to turn away from Me and they will worship the gods of others." From this, it is assumed the wife in this relationship is Jewish. There is no such statement regarding a child of a non-Jewish wife, inferring that the child is already a non-Jew, so there's nothing to be turned away from. In Ezra 10:2-3, when the Jews were returning to Israel after being exiled to Babylonia, they are told by Ezra to put away their "foreign" (non-Jewish) wives and any children born of them. The Shulchan Aruch ("The Set Table", published in 1563), is the Code of Jewish Law. It is a compilation of the decisions of rabbis throughout the ages based on the Talmud and other Jewish texts of law. Who a Jew may marry and the status of their offspring is set down in this text. It all boils down to what I said: Jewish mother-->offspring is Jewish. Non-Jewish mother-->offspring is not Jewish.