# Thread: Ring of Polynomials - and their evaluation - What is the link?

1. ## Ring of Polynomials - and their evaluation - What is the link?

Before I ask my question some standard definitions (or rather my understanding of these definitions):

Let $\displaystyle F[x]$ be a ring of polynomials over a field $\displaystyle F$.
i.e. $\displaystyle F[x]$ is a set of symbols of the form
$\displaystyle a_0x^0 + a_1x^1 + a_2x^2 + ... + a_nx^n$ where $\displaystyle a_i \in F$ and $\displaystyle x$ is just a symbol with no meaning whatsoever.

To make it into a ring we define + and . for $\displaystyle p(x),q(x) \in F[x]$. These definitions are nothing but syntax rules to manipulate these symbols (using the + and . which are already defined for the Field, $\displaystyle F$)

Under this definition we can write polynomial equations like
$\displaystyle p(x) = q(x).h(x) + r(x)$
Here all we mean is that when you apply the syntax rules for + and . defined on $\displaystyle F[x]$ you end up with same symbols on both sides of the equation. We assign no meaning to these symbols.

Now we define $\displaystyle x$ as an element of some algebric stucture $\displaystyle A$ where along with + and . we also define $\displaystyle \alpha.x$ where $\displaystyle \alpha \in F$ and $\displaystyle x \in A$. $\displaystyle \alpha.x \in A$. Under this, a polynomial evaluates into an element in the algebric structure $\displaystyle A$. (e.g. of such a structures are: field $\displaystyle F$ itself or a vector space $\displaystyle V$ over $\displaystyle F$)

My question is why should the LHS and RHS of a polynomial equation (as presented above and others as well) evaluate to a same element in $\displaystyle A$ ? What are the properties of $\displaystyle F[x]$ and/or $\displaystyle A$ that make this happen?

Not sure if this is relevant question or something too trivial. But I guess it is better to ask. Any pointers would be welcome.
Thanks

2. Originally Posted by aman_cc
Before I ask my question some standard definitions (or rather my understanding of these definitions):

Let $\displaystyle F[x]$ be a ring of polynomials over a field $\displaystyle F$.
i.e. $\displaystyle F[x]$ is a set of symbols of the form
$\displaystyle a_0x^0 + a_1x^1 + a_2x^2 + ... + a_nx^n$ where $\displaystyle a_i \in F$ and $\displaystyle x$ is just a symbol with no meaning whatsoever.

To make it into a ring we define + and . for $\displaystyle p(x),q(x) \in F[x]$. These definitions are nothing but syntax rules to manipulate these symbols (using the + and . which are already defined for the Field, $\displaystyle F$)

Under this definition we can write polynomial equations like
$\displaystyle p(x) = q(x).h(x) + r(x)$
Here all we mean is that when you apply the syntax rules for + and . defined on $\displaystyle F[x]$ you end up with same symbols on both sides of the equation. We assign no meaning to these symbols.

Now we define $\displaystyle x$ as an element of some algebric stucture $\displaystyle A$ where along with + and . we also define $\displaystyle \alpha.x$ where $\displaystyle \alpha \in F$ and $\displaystyle x \in A$. $\displaystyle \alpha.x \in A$. Under this, a polynomial evaluates into an element in the algebric structure $\displaystyle A$. (e.g. of such a structures are: field $\displaystyle F$ itself or a vector space $\displaystyle V$ over $\displaystyle F$)

My question is why should the LHS and RHS of a polynomial equation (as presented above and others as well) evaluate to a same element in $\displaystyle A$ ? What are the properties of $\displaystyle F[x]$ and/or $\displaystyle A$ that make this happen?

Not sure if this is relevant question or something too trivial. But I guess it is better to ask. Any pointers would be welcome.
Thanks
Requesting some help on this. Thanks