Is that $[\mathbb{C} : \mathbb{R}] = 2$ and is that $[\mathbb{C} : \mathbb{Q}]$ infinite?

Question

Since, $\mathbb{C} = \mathbb{R}(i)$ where $i = \sqrt{-1}$.
So, $[\mathbb{C} : \mathbb{R}] = [\mathbb{R}(i) : \mathbb{R}] = 2$.
Which $\mathbb{C}$ is a finite field extention of $\mathbb{R}$.

Since, there are infinite number of algebric numbers in $\mathbb{C}$.
So, $[\mathbb{C} : \mathbb{Q}]$ is infinite.
Which means $\mathbb{C}$ is an infinite field extention of $\mathbb{Q}$.

Are they correct ?

Answer 1

Remember that if $[F : K]=n\in\mathbb{N}$, $F$ is a vector space over $K$, with dimension $n$. So if $[\mathbb{R} : \mathbb{Q}]=n$, then $\mathbb{R}=\mathbb{Q}\times ... \times\mathbb{Q}$ $n$ times. That would imply that $\mathbb{R}$ is countable.

Thus,$[\mathbb{R} : \mathbb{Q}]=\infty$ and $[\mathbb{C} : \mathbb{Q}]=\infty$.