(3) You shall not intermarry with them: do not give your daughter to his son or take his daughter for your son.
(4) For he will turn your son away from Me and they will serve other gods, and Hashem's anger will blaze forth against you and He will promptly wipe you out.
This week's Parashas Vaeschanan brings proof to the world that a Jew's lineage is determined from his Jewish mother even she marries a gentile, and not from a Jewish father who marries a gentile. Note the Hebrew words carefully in Devarim 7:4.
"For he (the gentile father) will turn your (Jewish) (grand) son away from me." ~ Devarim 7:4
It does not say:
"For she (the gentile mother) will turn your (grand) son away from me."
Rashi says:
If the son of a gentile will marry your daughter, he (the gentile) will turn away your son (grandson) whom your daughter will bear for him, from following after Me. From this we learn that your daugther's son, who is born of a gentile, is considered your son; but your (Jewish) son's son, born of a gentile woman, is not considered your son, but rather her son; for it is not said "Do not take his daughter for your son, because she (the gentile) will turn [your (grand)son away from Me], but only that he (the gentile) will turn your (grand)son, etc.
Conclusion:
A child born of a Jewish mother and Gentile father is Jewish, but a child born of a Gentile mother and Jewish father is NOT Jewish.