Prove that ring contains infinitely many minimal prime ideals
Solution 1:
Write such primes by picking up one variable at a time from the products $x_1x_2$, $x_3x_4$, $x_5x_6$, and so on. (Every prime ideal containing $(x_1x_2,x_3x_4,x_5x_6,\dots)$ must contain one variable from each product.) For example, one of such minimal primes is $P=(x_1,x_4,x_5,\dots)$. It is clear (I hope) that in this way you will find infinitely many minimal primes.