TITLE: Do polynomials in two variables always factor in linear terms?
QUESTION [23 upvotes]: Consider a polynomial of one variable over $\Bbb C$:
$$p(x)=a_0+a_1x+\cdots+a_nx^n,\quad a_i\in\Bbb C.$$
We know from the Fundamental Theorem of Algebra that there exists $c,\alpha_i\in\Bbb C$ such that
$$p(x)=c(x-\alpha_1)\cdots(x-\alpha_n),$$
i.e. we can factor $p$ in linear terms.
Now, what about polynomials $p(x,y)$ in two variables? 

Is it still true that we can factor $p(x,y)$ in linear terms? I.e. can we always write
  $$p(x,y)=c(\alpha_1x+\beta_1y+\gamma_1)\cdots(\alpha_nx+\beta_ny+\gamma_n)$$
  for some $c,\alpha_i,\beta_i,\gamma_i\in\Bbb C$?

REPLY [7 votes]: To show that this is not the case, look at $x^2 + y^2 - 1 = 0$, which is true all along the unit circle, if your idea was true that we could factor down to "individual zero points", we would need to have infinite number of factors since there are infinitude of points on the unit circle. Also linear terms would not be possible, since the unit circle is curved and not a set of lines.
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