I am doing an exercise about investigating all the maximal ideals of the ring $\displaystyle \prod_{n=1}^{\infty} \mathbb{F}_2$. The question is that if there is any of them not of the form
$$\mathfrak{m}_{n}=\{(x_i)_{i \in \mathbb{N}}:x_n=0\}.$$
(Note that in this ring all the prime ideals are maximal, and of course the $\mathfrak{m}_n$'s are maximal)
I read this post before about the same problem but for an arbitrary field, not a finite field. From the first answer of this post I think that there could be a counter-example. I've tried many ways but all of them led to... non-prime ideals! I am reconsidering again if the statement is indeed true, but I'm not sure if it works. Any helps or hints from you guys are appreciated.