Basis for $\mathbb{[Q(\pi):Q]}$

Question

I'm trying to figure out whether the basis of $\mathbb{Q}(x)$ over $\mathbb{Q}$ is countable when $x$ is transcendental. I know that the elements in $\mathbb{Q}(x)$ will be rational functions in $x$ and so they are countable like algebraic numbers.

Let the rank be the sum of coefficients and degrees of polynomials in denominator and numerator.

So it passes the necessary condition for countable basis outlined by Asaf Karagila in Countable/uncountable basis of vector space. I have no more ideas to go with.

Edit:

Obviously a basis will have to be countable if the space is countable. So how do I go about finding a basis?

If there were an uncountable basis, how many distinct elements of $\mathbb{Q}(\pi)$ would there be? — Steven Stadnicki, May 07 '13 at 17:06
So, which part is your question. (1) $\mathbb Q(\pi)$ is countable, or (2) if $K \supset \mathbb Q$ is countable, then the dimension of $K$ over $\mathbb Q$ is (at most) coutable... — GEdgar, May 07 '13 at 17:07
Perhaps you can think of "partial fractions" to find a basis. — GEdgar, May 07 '13 at 17:09

score 6 · Accepted Answer · edited Jun 06 '23 at 21:14

I'll answer my own question using the suggestion by GEdgar in the comments. $\mathbb Q[x]$ is a Euclidean Domain and by the theorem of Partial Fractions

Let $f$ and $g$ be nonzero polynomials over a field $K$. Write $g$ as a product of powers of distinct irreducible polynomials :

$$g=\prod_{i=1}^k p_i^{n_i}$$

There are (unique) polynomials $b$ and $a_{ij}$ with $\deg(a_{ij})<\deg(p_i)$ such that

$$\frac{f}{g}=b+\sum_{i=1}^k\sum_{j=1}^{n_i}\frac{a_{ij}}{p_i^j}$$

If $\deg(f)<\deg(g)$, then $b=0$.

Let $$\mathcal{B}_1=\{x^n : n\in\mathbb{N}\}$$ and $$\mathcal{B}_2=\left\{\dfrac{x^n}{p^m} : p \text{ is a monic irreducible and } 0\le n < \deg(p)\right\}.$$

The theorem implies $\cal B_1 \cup B_2$ is a basis for $\mathbb Q(x)$ over $\mathbb Q$.

This is how this forum can work. We give some hints, and the inquirer finds a solution. — GEdgar, May 08 '13 at 13:14

Basis for $\mathbb{[Q(\pi):Q]}$

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