Shalom dear Rabbi,
Question: Why is only Jewish who is born of a Jewish mother is Jew, have a specific halakha, in the Torah explains both oral and written?
This is part of the oral torah, and the Talmud (Tractate Kiddushin 68b) explains this. It is written in the Torah, (Deuteronomy 7-3,4) “Don’t (inter)marry with them, your daughter don’t give to their son, and their son don’t take for your daughter. Lest he (the gentile son in law) make your (grand)son stray from before me and worship other G-ds…”. The gemora notes that it says that he, meaning your son in law- that is married to your (Jewish) daughter’s child will be let astray, and the Torah does not say, that your gentile daughter in law, (married to your son) will lead you grandchild astray? Why is this? Says the gemora we see from this, because the son of a gentile woman cannot be led astray because he isn’t Jewish! However, the son of your daughter, who is married to a gentile, who is still Jewish, can be led astray. This is because the Jewishness of a Jew depends on who the child’s mother is, not the father.