Prove or Disprove:
- $\Bbb F_2[x]$ has uncountably many maximal ideals
- For every integer $n$, every ideal of $\Bbb F_2[x]$ has only finitely many elements of degree $\leq n$.
The first one is false, since any maximal ideal is generated by an irreducible polynomial over $\Bbb F_2$ and the number $N_n$ of monic irreducible polynomials in $\Bbb F_2[x]$ of degree $n$ is $$N_n=\frac{1}{n} \sum_{d \vert n} \mu(d) \cdot2^{\frac{n}{d}}$$ where $\mu$ is a mobius function. It is a finite number for any given $n$. Let $A_n$ be the set containing possible monic irreducible polynomials of degree $n$. For example, $A_2=\{x^2+x+1\}$. Since each $A_i$ is countable and $$\text{number of maximal ideals}=\cup_n A_n$$ which is also countable. Thus first bullet is false
Is this correct? Any hint for the second one?
Best Answer
Regarding your first question: It seems you got the right idea, althought it's not really formally correct to write "number of maximal ideals = $\bigcup_n A_n$" (the left-hand side is a number, the right-hand side is a set.. etc.).
Regarding the second questions:
What does an element of degree $\leq n$ look like?
Assume the statement would be true. Then it would also be true in the case of the ideal $I = \mathbb{F}_2[X]$. And since $J \subseteq \mathbb{F}_2[X]$ for any ideal $J$ in $\mathbb{F}_2[X]$, the statement is true if and only if it is true for $\mathbb F_2 [X]$.
Hence what you really want to think about is:
How many polynomials $p \in \mathbb F_2[X]$ with $\deg p \leq n$ do exist for fixed $n \in \mathbb N$?