Does the theory of polynomials say something about their coefficients?

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SUMMARY

The discussion centers on the classification of polynomials based on their coefficients, specifically addressing whether polynomials with all imaginary coefficients can still be considered "normal." It is established that polynomials can exist over various mathematical objects, including real numbers, complex numbers, integers mod 17, and matrices. The term "normal" is context-dependent, with real and complex polynomials being the most common forms, while others are categorized as more exotic.

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theriel
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Hello! I have just a small question - does the theory of polynomials say something about their coefficients? I mean: is the polynomial with all the coefficients being imaginary still considered as a "normal' polynomial?
 
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In some sense, it's possible to have polynomials over any mathematical object where multiplication and addition is defined. So, for example, one might have a polynomial over the real numbers (a.k.a. a real polynomial), over the complex numbers (a complex polynomial), the integers mod 17, or 7x7 matrices. When in doubt, it is always safe to specify what type of object is being operated with.

The notion of "normal" is basically a result of context. Real and complex polynomials are both very common. The others are more exotic.
 

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