# How is an arbitrary member of $\mathbb{Q}(\pi)$ defined?

How is arbitrary member of $$\mathbb{Q}(\pi)$$ defined? $$\mathbb{Q}(\pi)$$ means the extension of $$\mathbb{Q}$$ by $$\pi$$ , thanks

I thought maybe it's

$$x+y\pi,\quad x,y\in\mathbb{Q}$$

but how to get $$\pi^2=x+y\pi$$ or $$1/\pi$$

with that?

• Since $\pi$ is transcendental, $\Bbb Q (\pi)$ is an infinite extension. So I think your approach is wrong. – Thomas Shelby Mar 19 at 0:18
• It simply comprises all rational fractions in $\pi$ with integer coefficients. – Bernard Mar 19 at 0:20
• @Bernard can you give examples of this please? in there a general expression? – Loli Mar 19 at 0:38
• Do you have any reason to believe that $\pi^2$ and $\frac{1}{\pi}$ are in Q(π)? – user247327 Mar 19 at 0:48
• @Thomas Shelby I think a typical element is $p(\pi) / q(\pi)$, $p, q \in Q[x]$ – Loli Mar 19 at 1:38

$$\Bbb Q(x)$$ is isomorphic to $$\Bbb Q(π)$$, by sending $$x$$ to $$π$$. So, no typical element is of the form $$x+πy$$, as that'd mean we are dealing with a degree $$2$$ extension, but $$\Bbb Q(π)$$ is an infinite extension. This is always the case when you adjoin a transcedental number to $$\Bbb Q$$.

• make sense! thanks – Loli Mar 19 at 1:09
• I think a typical element is $p(\pi) / q(\pi)$, $p, q \in Q[x]$ – Loli Mar 19 at 1:37