# Find the $[\Bbb{R} ^{\text{alg}}:\Bbb{Q}]$

If we have the field extension $\Bbb{R}:\Bbb{Q}$, we define the set $$\Bbb{R} ^{\text{alg}}:=\{a\in \Bbb{R}: a \text{ is algebraic over }\Bbb{Q} \}$$ We want to show that $\Bbb{R}^{\text{alg}}$ is a subfield of $\Bbb{R}$ and find the $[\Bbb{R} ^{\text{alg}}:\Bbb{Q}]$.

If we consider that $a,b\in \Bbb{R} ^{\text{alg}} \iff a,b$ are algebraic over $\Bbb{Q}$, we have that the extension $\Bbb{Q}(a,b):\Bbb{Q}$ is algebraic over $\Bbb{Q}$. We know that $\Bbb{Q}(a,b)$ is a field, so $a-b\in \Bbb{Q}(a,b)$ and if $b\neq 0,\ ab^{-1} \in \Bbb{Q}(a,b)$, and from the previous observation are algebraic. So, $a-b,\ ab^{-1} \in \Bbb{R}^{\text{alg}}$. And this means $\Bbb{R}^{\text{alg}} \leq\mathbb{R}$.
Right? And how can we find the $[\Bbb{R} ^{\text{alg}}:\Bbb{Q}]$?
• Do you have any guesses about what $[\Bbb R^{\rm alg}:\Bbb Q]$ might be? – arctic tern Apr 6 '17 at 0:08
• @arctictern Thank you for your comment. Maybe $\infty$? But I can't find a way to determine a basis! – Chris Apr 6 '17 at 0:13