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Historically the complex numbers have been treated as a mysterious object, due to there being polynomials over the real field having no real roots. Thus mathematicians were inclined to define roots of these polynomials as if they existed on some space outside the real field. But many standard algebraic operations such as exponentiation and logarithms do not work the same way they do on complex numbers as they do with reals.

In one sense, the complex number is an algebraic object that allows us to form an algebraically closed field, that is every polynomial over the complex field must split over the complex field. This has important applications in linear algebra, such as guaranteeing the existence of a Jordan normal form for matrices over the complex field.

Is it is a theorem of algebra that the complex field cannot be ordered? Is it a theorem of algebra that the complex field is the smallest algebraically closed field containing the reals? What fields/vector spaces is the complex field isomorphic to?

And how are the complex numbers really axiomatized? As algebraic objects, or as geometric objects?

The reason I ask these questions is that complex numbers are immensely useful in the field of signal processing, but because the field is algebraically closed, but more because trigonometric functions can be written in the form of complex exponentials and that computations on complex exponentials are much faster. Most of the use of complex numbers in signal processing is from the Euler identity combined with the Fourier transform.

So far, most textbooks I have seen define complex numbers as a two dimensional vector space over the reals with an embedded product such that (0,1)*(0,1) = (-1,0). This geometric interpretation is handy, as it reduces the mystery of the complex field to a vector space isomorphic to ##ℝ^{2}## with some additional structure, but if this is the direction we take, how can we develop the link between such a geometric interpretation and the inevitable powerful algebraic properties of the complex field and reconcile the two?

And then there's Euler's identity, and calculus on complex-valued functions. How does that work?

A recommendation on a textbook, but more importantly, a brief explanation (assuming one exists) in your own words that may resolve my questions is deeply appreciated!

BiP