What is the index of field extension $\mathbb{C}/\mathbb{R}$?

Question

What is the index of field extension $\mathbb{C}/\mathbb{R}$?

I know that the answer is $2$, but if so, that means $\mathbb{C}/\mathbb{R}=\{\mathbb{R}, i+\mathbb{R}\}$, and how come $5i+\mathbb{R}=i+\mathbb{R}$?

Answer 1

The degree of the field extension is 2: $[\mathbb{C}:\mathbb{R}] = 2$ because that is the dimension of a basis of $\mathbb{C}$ over $\mathbb{R}$.

As additive groups, $\mathbb{R}$ is normal in $\mathbb{C}$, so we get that $\mathbb{C} / \mathbb{R}$ is a group. The cardinality of this group is uncountably infinite (we have an answer for this here), which you should attempt to prove along the lines you suggested in the body of your question.

Answer 2

Take the map $f : \Bbb C \longrightarrow \Bbb C$ defined by $a+bi \mapsto bi,\ a,b \in \Bbb R.$ Observe that the kernel of the map $f$ is isomorphic to $\Bbb R$ and the image is the imaginary axis. So by the first group isomorphism theorem we can conclude that as a group $\Bbb C/\Bbb R$ is isomorphic to the imaginary axis which is uncountably infinite.