Question about field extension notation

Question

Hello all I was given the following question about which I understand everything except possibly the notation. I am given a sub-field $ F \subseteq R $ and I am asked to prove the degree of the "field extension" $ F[i] / F $ is two. My problem is that everywhere else in the assignment they used circular brackets while in this they used square brackets. They did refer to this as a field extension. Could you please explain this notation? Thank you all

Answer 1

In this case $F[i]=F(i)$. This is because $F(i)$ is the field of fractions of $F[i]$, which is already a field. You can see this becuase the characteristic polynomial of $i$ is irreducible in any subfield or $\mathbb{R}$, or more explicitly by computing $(a+bi)\frac{1}{a^2+b^2}(a-bi)=1$, so that the inverse of any nonzero element exists.