Let $K$ be a field and consider the ring $A=K[x_1,x_2,\cdots]$ in countably infinite indeterminates and its ideal $\alpha=(x_1,x_2^2,\cdots,x_n^n,\cdots)$. Then Atiya and MacDonald in their "Introduction to Commutative Algebra", top of page 91, mention that the only prime ideal of $A/ \alpha$ is the image of $(x_1,x_2,\cdots)$. I can see that the image of this ideal is prime, however how can we see that it is the only prime?
Thanks.