Binary Polynomial Cardinality

For it to be countable there has to exist a bijection (one-to-one and onto) function between the natural numbers and the set of polynomials modulo 2.

Firstly consider polynomials in modulo 2. We have ones like $x^2+x+1$ and $x^3+1$ . We either have a term of a certain degree or we do not. Therefore we will employ binary notation to represent polynomials. $1101=x^3+x^2+1$ .

Now that we have a notation for polynomials in $\Bbb Z_2[x]$ we can establish our bijection.

Let $\phi:\Bbb N\to \Bbb N$ be our desired bijection. We want to show that for a $k$ we have a corresponding set of polynomials. Namely $\phi:k\mapsto 2^{k-1}$ describes our counting. Let me demonstrate:

We have $k$ the degree of polynomials to count from downwards. So we count all polynomials with degree $k$ and below.

Clearly $\phi$ is counting the number of polynomials and is a bijection between $\Bbb N$ therefore $|\Bbb Z_2[x]|=|\Bbb N|=\boxed{\aleph_0}$