On algebraic numbers

Question

Walter Rudin Exercise 2.2

To prove that the set of all algebraic numbers is countable, the hint provided is that there are finitely many equations of the form $$n+\left|a_0\right|+\left|a_1\right|+\dots+\left|a_n\right|=N. $$ where $a_i$ are coefficients, $n$ is degree of polynomial and $N$ is any positive integer.

How is this analogous to the problem?

Answer 1

Show that for any $n\geqslant 1$, the set of roots of polynomials with integer coefficients of degree exactly $n$ is countable, then use that a countable union of countable sets is countable.

Answer 2

I'm not entirely sure of what Rudin meant with the hint, but the general spirit of his approach is that we can separate algebraic numbers in terms of which degree the polynomials they are roots of have, and then separate the polynomials of a given degree in terms of their coefficients. All these will be countable, so when we take the union of them the result will remain countable.

Concretely, let's first see that $\mathbb{Z}_n[X]$ the integers polynomials of grade $n$ are countable: note that the function that maps $a_nX^n + \dots + a_0$ to $(a_n, \dots, a_0) \in \mathbb{Z}^n$ is injective. Furthermore, since

$$ \mathbb{Z}[X] = \bigcup_{n \geq 0}\mathbb{Z}_n[X] \cup \{0\} $$

we have that $\mathbb{Z}[X]$ itself is countable. Finally, the algebraic numbers can be written as the union of the roots of all polynomials with integer coefficients:

$$ \mathbb{A} = \bigcup_{p \in \mathbb{Z}[X] \setminus \{0\}}p^{-1}(0) $$

Each preimage $p^{-1}(0)$ is finite, because $p$ has finite degree, and the union is countable because $\mathbb{Z}[X]$ is, so the result is countable itself.

Answer 3

An attempt.

Consider:

$a_0 z^n+ a_1z^{n-1}+...+a_{n -1}z+a_n=0$

where $a_{i}$ , $i=0,1,2,.....n,$ are integers, $a_0\not =0$.

To every polynomial of degree $n$ corresponds a $n$-tuple

$ (a_0,a_1,........, a_{n-1},a_n)$ (bijection).

The set

$T_n:=$

{$(a_0,a_1,...,a_{n-1},a_n)|a_0 \not =0$}

$\subset \mathbb{Z^{n+1}}$.

As a subset of a countable set,

$T_n$ is countable.

Let $t_{kn} \in T_n, k=1,2,.....$

Every $t_{kn}$ corresponds to a polynomial of degree $n$ which has $n$ roots.

Arrange the roots in dictionary order: $z_1,z_2,.......z_n$.

$ A_n:= ${$(t_{nk}, j)| 1\le j \le n$, $j$ integer, $k=1,2,...$}

is as subset of $T_n×\mathbb{N}$ countable .

Finally :

$\cup_{n} A_n$, countable union of countable sets, is countable.