TITLE: A question based on ring $\mathbb{F}_{2}$ [X]
QUESTION [0 upvotes]: This question was asked in my abstract algebra quiz and I was unable to solve it.

Let $R = \mathbb{F}_{2}$ [X] . Then choose the correct option(s):
$1$. $R$ has uncountably many maximal ideals.
$2$. Every maximal ideal of $R$ has infinitely many elements.
$3$. For all maximal ideals, $m$ of $R$, $~R/m$ is a finite field.
$4$. For every integer $n$, every ideal of $R$ has only finitely many elements of degree $\leqslant n$.

Although I am not able to prove/ disprove any of the options kindly just tell me the reasoning behind only 1st options. Rest I would like to work by myself. If unable to do ask, then I will ask here later.

REPLY [0 votes]: The ring of polynomials over a field is Euclidean, and thus a PID. So every ideal (including all maximal ideals) are of the form $(p)$, where $p$ is a polynomial in $\mathbb F_2[X]$. In particular, there are at most as many maximal ideals as there are polynomials in the first place. Can you show that their number is countable?
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