Commentary for Kiddushin 136:3

The Talmud now explains how we know that kiddushin is invalid with a non-Jewish woman and that her offspring is not Jewish. First of all, the Torah says “Do not marry them.”
As to the second question, the verse warns that if a non-Jewish man marries a Jewish woman, the son will be turned away from God. Thus the child of a Jewish woman is called “your son” meaning an Israelite. By deduction, we can conclude that the child of a Gentile woman is not your son.
Note that all of these explanations are clearly “derashot”—interpretations of the verse meant to accord with known halakhah. As I said in my intro to the mishnah, in the Torah there is little doubt that genealogy goes through the father. None of these verses actually means that it goes through the mother. This is the rabbis reading their own halakhah into these verses. Indeed, later in this passage, the Talmud will basically admit that.