A complex number over a field F is defined as a+bi where a,b∈F and i is the square root of the inverse of the multiplicative identity of F, denoted as i2=−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 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=α+βi, we can define the square root of c as a complex number a+bi st. a2−b2=α,2ab=β. But I don't know if this definition is appropriate.
Any help and reference is appreciated. Thank you!
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