Does the theory of polynomials say something about their coefficients?

[theriel]

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?

 

[NateTG]

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.

 

[tacman]

Yes, the theory of polynomials does say something about their coefficients. In fact, the coefficients of a polynomial play a crucial role in determining its properties and behavior. The degree of a polynomial is determined by the highest power of the variable, while the coefficients determine the specific values of the polynomial at different points. Additionally, the coefficients also determine the roots of the polynomial, which are the values of the variable that make the polynomial equal to zero.

In regards to your question about a polynomial with all imaginary coefficients, it would still be considered a "normal" polynomial. The coefficients of a polynomial can be any real or complex numbers, so having all imaginary coefficients does not make it any less of a polynomial. However, in some cases, having all imaginary coefficients may result in a polynomial with no real roots, which can make it more difficult to solve or analyze. But overall, the theory of polynomials applies to all types of coefficients, including imaginary ones.