Who is a Jew?

We Were Wrong to Assume Ella Emhoff Is Jewish, by Molly Tolsky in Hey Alma
In an article published last Friday, Forward editor-in-chief Jodi Rudoren reported that when reached out to participate in an event for the Jewish newspaper, a spokesperson for Ella declined, saying, “Ella is not Jewish.” In subsequent emails, the spokesperson clarified that while Doug Emhoff had started celebrating Judaism “out of an independent search” over the last few years, Ella was not living with him at the time and Judaism is “not something she grew up with.” For that reason, “Ella truly has no qualms with the faith, but she does not want to speak on behalf of Judaism, as she does not celebrate herself.”

Bobby Fischer letter to Encyclopedia Judaica Jerusalem, June 28, 1984
Knowing what I do about Judaism (sic), I was naturally distressed to see that you have erroneously featured me as a Jew in ENCYCLOPAEDIA JUDAICA. Please do not make this mistake again in any future additions of your voluminous, pseudo-authoritative publication. I am not today, nor have I ever been a Jew, and as a matter of fact, I am uncircumcised.

I suggest rather than fraudulently misrepresenting me to be a Jew, and dishonestly abusing my name and reputation as a kind of advertising gimmick to improve the image of your religion (Judaism), you try to promote your religion on its own merits—if indeed it has any!

Tablet Magazine paraphrasing A Bobby Fischer radio Broadcast in the Phillipines
The Jews are a "filthy, lying bastard people" bent on world domination through such insidious schemes as the Holocaust ("a money-making invention"), the mass murder of Christian children ("their blood is used for black-magic ceremonies"), and junk food (William Rosenberg, the founder of Dunkin' Donuts, is singled out as a culprit).

Biblical Characters that are Patrilineal:

• The Children of Yoseph (married an Egyptian) - Manassah & Ephraim (granted the honor of being descendants of Moshiach Yoseph)
• The Children of Moishe and Tziporrah (Midian) - Gershom and Eliezer

Exert from Chabad.org Article: Why Is Jewishness Matrilineal? Maternal Descent In Judaism By Tzvi Freeman and Yehuda Shurpin
Ezra is one of the last books of the Hebrew Bible. Many have asked why this issue was not raised earlier—even in the times of Moses?
If the Hebrew Bible provided a detailed exposition of every law and custom, this would be a question. But with even a cursory look it’s obvious that this is not so. Some laws, such as priestly rites and offerings, or the design and artifacts of the Tabernacle are described in detail.
Others—generally the most common ones—are presented with no specifics at all.
For example, we are told to “rest on the seventh day” and “do no work.” What is rest? Are we meant to sleep the entire day? What is work? Does it mean exertion, or productivity, or just anything not enjoyable? The few details that the text provides are of little help. (“Don’t burn fire in your dwellings.” “No one should move from their place.”)
Moses is told that we are to slaughter an animal “… as I commanded you.” But nowhere are we provided the details of just what Moses was instructed.
So why are some details provided and others left out?
One possible answer is that it is only necessary to commit details to writing that are easily forgotten. Those matters that all the people are intimately familiar with are left to be passed down organically and tacitly.
But whatever the reason, the fact that there is no explicit verse stating which parent makes you Jewish is not unusual.
There are many things in Jewish tradition that are hard to understand. The Five Books of Moses provide many instructions, but rarely provides the “why.” And even the explanations that we do have are not the ultimate reason, as Rabbi Schneur Zalman of Liadi explains. We’re speaking, after all, of the wisdom of our Creator. It’s wondrous that we understand anything at all.
Nevertheless, in this instance, writes the Rebbe, Rabbi Menachem M. Schneerson, of righteous memory, not only is matrilineal descent perfectly reasonable, it’s absurd to imagine otherwise.
Let’s start with a simple question: To which parent is the child more biologically connected?
Yes, the father provides half the chromosomes. But there’s more to a person than two sets of chromosomes.
An embryo must sit within its mother’s womb and develop for nine months. During those months, the fetus is nurtured from the bloodstream of the mother, affected by her emotions, by the sounds she hears and the places she goes. And none of this, especially the birthing process, is a terribly comfortable experience for the mother.
This aside from the nursing and nurture, both physical and psychological, in the primary years that are most critical to the child’s development. Generally, that’s provided by the mother, who is capable of providing far more than the man.
Now, ask yourself, to whom would you give dominance in the fundamental identity of this child? Is it justified to ignore the woman’s role, and identify the child after the father? If the mother is, say, Cherokee, determining that the child is Jewish because the father is Jewish is in conflict with the basics of biology.
The Rebbe sharply criticized those who insist we change Jewish law and accept patrilineal descent. Firstly, he said, they cannot succeed. Torah law does not change, and every attempt to make changes of this sort has failed in the long run.
But aside from that, these people do not realize what they are playing with. Each human being is created with a role in this world that is tied to their identity. To tell a person that he is Jewish when he is not is a crime against both mother and child. It is robbery in the worst sense to steal away a person’s identity and the fundamental meaning of their life.
That’s a point many who argue for patrilineal descent ignore. They argue that we must have compassion, that we must be more inclusive, and we should open the doors of Jewishness to those who have either a mother or father who is Jewish. Why demand a conversion in these cases?
But this argument ignores not only the fundamental makeup of the child, but the dignity of the mother of the child, as well as women in general. “I would have expected all the women of the world to protest,” the Rebbe said.
And it is extremely ethnocentric, as though Jewish identity exists in a vacuum. There are people in the world besides Jews, and they also have identities. By simple biological fact, that identity principally follows the mother.

Exert from the 1983 Central Conference of American Rabbis report entitled "Report of the Committee on Patrilineal Descent" which made it reform practice to accept patrilineal Jews
Jewish law recognizes a person as Jewish if his mother was Jewish, even though the father was not a Jew. One born of such mixed parentage may be admitted to membership in the synagogue and enter into a marital relationship with a Jew, provided he has not been reared in or formally admitted into some other faith. The child of a Jewish father and a non- Jewish mother, accoridng to traditional law, is a Gentile; such a person would have to be formally converted in order to marry a Jew or become a synagogue member.
Reform Judaism, however, accepts such a child as Jewish without a formal conversion, if he attends a Jewish school and follows a course of studies leading to Confirmation. Such procedure is regarded as sufficient evidence that the parents and the child himself intend that he shall live as a Jew. (Rabbi's Manual, p. 112)
We face today an unprecedented situation due to the changed conditions in which decisions concerning the status of the child of a mixed marrige are to be made.
There are tens of thousands of mixed marriages. In a vast majority of these cases the non-Jewish extended family is a functioning part of the child's world, and may be decisive in shaping the life of the child. It can no longer be assumed a priori, therefore, that the child of a Jewish mother will be Jewish any more than that the child of a non-Jewish mother will not be.
This leads us to the conclusion that the same requirements must be applied to establish the status of a child of a mixed marriage, regardless of whether the mother or the father is Jewish.
Therefore:
The Central Conference of American Rabbis declares that the child of one Jewish parent is under the presumption of Jewish descent. This presumption of the Jewish status of the offspring of any mixed marriage is to be established through appropriate and timely public and formal acts of identification with the Jewish faith and people. The performance of these mitzvot serves to commit those who participate in them, both parent and child, to Jewish life.
Depending on circumstances, mitzvot leading toward a positive and exclusive Jewish identity will include entry into the covenant, acquisition of a Hebrew name, Torah study, Bar/Bat Mitzvah, and Kabbalat Torah (Confirmation). For those beyond childhood claiming Jewish identity, other public acts or declarations may be added or substituted after consultation with their rabbi.