Domain of a polynomial

By default, we think of polynomials over the real numbers or complex numbers.

However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.

For example, given the finite field of order 9, $GP(3)$ and with elements $\{0,1,2\}$, we can denote polynomials over that ring as where $x$ is the variable name.

For example, one such polynomial could be: and another one: Note how all the coefficients are members of the finite field we chose.

Given this, we could evaluate the polynomial for any element of the field, e.g.: and so on.

We can also add polynomials as usual over the field: and multiplication works analogously.

The usual definition of a polynomial is over a field as shown at polynomial over a field.

However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.

The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms: while for a field they would all go into a single term: so when considering a polynomial over a ring we end up with a lot more more possible terms.

If the ring is a commutative ring however, polynomials do look like proper polynomials: Section "Polynomial over a commutative ring".

Unlike over non-commutative rings, polynomials do look like proper polynomials over commutative ring.

In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.