Monday, November 27, 2017

abstract algebra - Is the square root of a complex number over a field always well-defined?

A complex number over a field F is defined as a+bi where a,bF 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.





  1. 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?


  2. 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. a2b2=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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