A complex number over a field $F$ is defined as $a+b\text{i}$ where $a,b\in F$ and $\text{i}$ is the square root of the inverse of the multiplicative identity of $F$, denoted as $\text{i}^2 = -1$. I have several questions about this definition.
Is there any restriction on $F$ in addition to that the square roots of $-1$ must exist? Should $F$ be a quadratically closed field, i.e. every number in $F$ has a square root?
How square root of a complex number is defined? The square root of a complex number over $\Bbb R$ can be defined by De Moivre's formula, but I don't see how this is extended to an arbitrary field.
For question 2, I can imagine a simple definition. For a complex number $c=\alpha+\beta\text{i}$, we can define the square root of $c$ as a complex number $a+b\text{i}$ st. $a^2-b^2=\alpha,2ab=\beta $. But I don't know if this definition is appropriate.
Any help and reference is appreciated. Thank you!
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