Multiple textbooks that I've looked at make some comment about how the fact that the evaluation map $\phi_c: F[x] \rightarrow F$ is a homomorphism "allows us" to evaluate the polynomial at a specific $c \in F$. However I'm having trouble understanding why it is important that this map be a homomorphism, or even why we need the map at all. Without it, what stops us from just considering, for any polynomial $a_nx^n + \dots + a_1x + a_0$ in $F[x]$, the element $a_n c^n + \dots + a_1 c + a_0$ in $F$? Why do we need the theorem that the evaluation map is a ring homomorphism?
Ring Theory – Importance of Evaluation Map as a Homomorphism
field-theorygroup-homomorphismpolynomialsring-theory
Best Answer
TL;DR: An essential property of polynomials is that they can be associated to* functions on their coefficient rings. The fact that the evaluation map is a homomorphism is what makes this possible.
In school, one is often taught that polynomials are expressions in which we can "plug in" numbers and "everything works out the same". The evaluation map being a ring homomorphism is how one takes this informal, intuitive understanding of polynomials and makes it rigorous and formal.
*(not necessarily in a bijective manner, see this question)
In some sense, the fact that the evaluation map is a ring homomorphism is the defining property of a polynomial ring. Such a property is also called a universal property.
UNIVERSAL PROPERTY OF POLYNOMIAL RINGS (See also)
In other words, the fact that the evaluation map is a ring homomorphism is the very reason to even bother thinking about polynomial rings in the first place.
The evaluation map being a homomorphism means that it is possible to associate abstract polynomials with actual functions on the ring. If the evaluation map were not a homomorphism, multiplication of polynomials would not correspond to multiplication of their "values" at every point of the ring, and addition of polynomials would not correspond to the addition of their "values" at every point of the ring. Since we expect multiplication and addition of functions to be defined pointwise, the evaluation map being a homomorphism is a necessary condition for the elements of the polynomial ring to correspond to functions on the coefficient ring. (It is also sufficient.)
For example, see this quote from the above-linked page:
So perhaps one could say that the book you are reading has the order backwards: instead of defining polynomial rings in terms of the universal property that evaluation maps are ring homomorphisms, they have defined a polynomial ring in terms of a specific construction, and then showed that this construction satisfies the universal property.
This is an understandable approach, because defining something in terms of universal properties may not be a good idea if no structure actually exists which satisfies the universal property.
Still, thinking in terms of universal properties, and subsequently proving that a structure exists which satisfies the universal property, may be more straightforward to some people.
Thinking in terms of universal properties first and constructions second is somewhat akin to declarative programming: we "declare" what properties we want a certain object to have, without worrying as much about the "implementation" (how to construct an object satisfying the property).
The approach used by your textbook is somewhat more like imperative programming: we specify how to construct an object, therefore being assured that it exists, and then retroactively examine what properties this constructed object has.
In either case, we have to worry about both whether an "implementation" exists as well as what properties we want, but in some sense we have the liberty to choose which concern to address first, or at least worry about more, depending upon our preferences and/or personal strengths.