Describing the Extension Field $\mathbb{Q}(\pi)$ of $\mathbb{Q}$

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Problem 81 (Describing the Extension Field $\mathbb{Q}(\pi)$ of $\mathbb{Q}$)

Consider the field $\mathbb{Q}$ and the element $\pi$. It is known that the element $\pi$ is not the zero of any polynomial in $\mathbb{Q}[x]$. However, we also know that $\pi\in \mathbb{R}$, which is an extension field of $\mathbb{Q}$. This means we can talk about $\mathbb{Q}(\pi)$, the smallest subfield of $\mathbb{R}$ that contains both $\mathbb{Q}$ and $\pi$. Describe the elements of $\mathbb{Q}$ in a constructive way. In other words, give a way to obtain all elements of $\mathbb{Q}(\pi)$ by giving a way to describe any element of this field.

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