Invisible wrote:
>>> It is not immediately clear to me that the quotient of two 
>>> polynomials is necessarily a polynomial.
>>
>> Nope, same as dividing two integers doesn't always give an integer 
>> answer, you sometimes end up with fractional parts in the answer.
> 
> Right. So there should also be a polymonial modulus operator?

Indeed there is.  It is a useful operation in abstract algebra, 
particularly in extending sets of numbers so that they contain elements 
with useful algebraic properties.  For instance, if you consider 
polynomials over the real numbers modulo X^2-1, you get the same 
structure as the complex numbers (you can do a similar thing for 
quaternions).

In terms of real-world uses, iirc the polynomial modulus is valuable in 
in information coding.  For example modular multiplication of 
polynomials is a step in the AES cypher and a CRC checksum is 
essentially a polynomial modulus.

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