Why are polynomials defined the way they are in algebra?

[Mr Davis 97]

I've always been curious about why we define polynomials the way we do. On the surface, it seems that they are expressions that naturally arise from combining the standard arithmetic operations on indeterminates. However, there are some points that I am generally confused about. Why are polynomials defined to have only powers of $x$ that are non-negative integers? Why not rational or negative powers? Also, in studying polynomials, they invariably have coefficients in $\mathbb{Q}$. Why do we seem to be not as concerned with polynomials with coefficients in $\mathbb{R}$ or $\mathbb{C}$? For example, in abstract algebra in studying extension fields, the coefficients of polynomials under study are most always assumed to be in $\mathbb{Q}$, with $\mathbb{R}$ or $\mathbb{C}$ as the extension fields of $\mathbb{Q}$. Also, the ideas of transcendental and algebraic numbers seem to be defined in terms of polynomials with coefficients in $\mathbb{Q}$

 

[jedishrfu]

We relate complicated functions in terms of polynomials to make it easier to compute answers.

By using powers that are beyond natural numbers complicates the calculations immensely similarly for the coefficients. We strive to use simpler constructs to understand more complicated constructs in math.

An analogous example, is the use of compass and straight edge in geometry versus using a protractor and a ruler. Clearly you can do more with a protractor and ruler but it's not as precise as doing it with a compass and straight edge.

For those problems where compass and straight edge don't work then we search for a new idea like using origami as was recently shown the solve the trisectionmof an angle and the doubling of a cube.

A more detailed answer to your question may be found here:

https://math.stackexchange.com/questions/579783/taylor-polynomials-why-only-integer-powers

 

[fresh_42]

In algebraic fields like group, ring or field theory, one is interested in whether an element $a$ can be found in those structures or not. If yes, then it satisfies an equation made of the basic operations allowed, a zero of a polynomial. If no, we ask, whether there isn't such an equation at all or in some extension domain, in which case we distinguish between transcendental or algebraic elements. We cannot ignore the fact, that there is a fundamental difference between $\pi$ and $\sqrt{2}$. The more as some interesting historical questions depend on it. If an element can only be viewed as a formal extension, we have constructions for them as well, e.g. group algebras. But as you've already mentioned, within the possibilities of the given operations.

The restriction to the fields you named isn't quite correct. One basic distinction is for example the characteristic of a domain. And $\mathbb{Q}$ simply happens to be the prime field of fields of characteristic $0$. But others are investigated as well. The situations you mentioned all belong to the fundamental properties of algebraic structures. It doesn't end here, it's simply the basic principles.

The study of polynomials over real or complex numbers, the extension to other powers than natural numbers are all treated by analysis and build a cosmos on their own. The emphasis here aren't individual elements anymore, but the general behavior of functions, i.e. a variety of elements. There are several concepts studied, which can be settled in the various realms between analysis and algebra: algebraic geometry, quadratic forms, various algebras, analytic manifolds, commutative algebra and many which depend partially on constructions introduced in algebra, like e.g. measure algebras in measure theory.

Your question is a bit like: Why do we have to learn linear algebra? Why in the world do we restrict ourselves on polynomials of degree $1$ which haven't even a constant term? And even more, there isn't hardly anything out there, that is really linear!