When did the rabbis start ruling that your mother has to be Jewish for you to be a Jew? by Yair Davidiy (21 May 2017, 25 Iyar, 5777)
Judaism says that if your mother is Jewish then so are you. This is a general rule. In all individual cases it is recommended that an Orthodox Rabbi be consulted.
This law is based on the Bible as shown below.
Even though it is derived from the Bible it could theoretically have been a later development since the Rabbinical Sages had the authority of Scripture to make such decisions, see Deuteronomy 17:8-13, and other sources.
The Talmudic source is Mishnah Kiddushin 3;12 see Gemara Kiddushin 68b.
The ruling is consistent with the Hebrew text and internal Biblical evidence suggests that this was the law from the beginning.Â
Why not accept that it was? Especially in lieu of contrary evidence?
In Ezra 10:3 we find foreign women and their children by Jews being sent away.
Ezra 10:
3 NOW THEREFORE LET US MAKE A COVENANT WITH OUR GOD TO PUT
AWAY ALL THE WIVES, AND SUCH AS ARE BORN OF THEM, ACCORDING TO THE COUNSEL
OF MY LORD, AND OF THOSE THAT TREMBLE AT THE COMMANDMENT OF OUR GOD; AND LET
IT BE DONE ACCORDING TO THE LAW.
This indicates that the disallowance of non-Israelite women and their children preceded Ezra.
Ezra shows that descent from foreign women (without conversion as in the case of Ruth) was not accepted.
It could be argued that Ezra may also have rejected the offspring of Jewish women in the same way as he did those of Gentile females.
How do we know that descent from Israelite women was then the rule?
In reply:
(1) We have the present ruling (tracing Jewish ethnicity through the mother) and it fits the deed of Ezra.
(2) There are problems of Exegesis (Biblical Commentary) that would be solved (as proposed by others) through accepting the ruling, e.g.
Zeruiah (sister of David) the mother of Abishai, Joab, and Asahel. Zeruiah is mentioned but her husband is not.
 1-Chronicles 2:
16 and their sisters were Zeruiah and Abigail. And the three sons of Zeruiah were Abishai, Joab and Asahel.Â
Suppose the husband of Zeruiah was not an Israelite. That would explain why only the mother is recalled in this case.
(3) The Hebrew Text supports the Rabbinical Explanation, as shown below.
Who is to say that such was not the practice from the beginning?
Here is the Biblical text::
Deuteronomy 7:
3 Nor shall you make marriages with them. You shall not give your daughter to his son, nor take his daughter for your son. 4 For he will turn your son away from following Me, to serve other gods; so the anger of the LORD will be aroused against you and destroy you suddenly.
The verse 7:3 says: Do not give your daughter to his son. Nor take his daughter for your son.
We have here two couplings:
(a) Your daughter and his son;
and (b) his daughter and your son.
The following verse (7:4) says:Â He will turn your son away from me. It only speaks of one of the two coupling. This is parallel to the first coupling (a) in 7:3 i.e. your daughter and his son.
The son who will be turned away in 7:4 is the son of your daughter and his son. He is YOUR son.
The second coupling (b) in 7:3 is not mentioned in 7:4 since the offspring is not your son.
The verse (7:3) says:
(a) Not to give your daughter to his son.
(b) Not to take his daughter for your son.
If you transgress the result will be (7:4) that:
(c) He will turn your son (i.e. your grandson) away from following God. "He" (who does the turning away) is the non-Israelite husband of your daughter. "Your son" is the offspring of your daughter.
Going through it again:
Let us call the son of the pagan "X".
The Grammatical construct in the Hebrew can be understood to say:
Your daughter you shall not give to "X". You shall not take his daughter [the sister of "X"] for your son [Deuteronomy 7:3] .
For "X" will cause your son to turn away from me [Deuteronomy 7:4]."
Who is "your son" that "X" will cause to turn away?
Answer: The son of your daughter that you have given to "X" and who "X" has begotten.
The son of your daughter by a Gentile is called "your son".
Incidentally, the Biblical Commentator, Abarbanel ("Mashmia Yishuah,"  2; 3), understood a Biblical text to indicate that in the End Times when the Lost Israelites return it will not be as it was in the time of Ezra.
He says that at that time the returnees will be allowed to keep their foreign wives and their children will be recognized.