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

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

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

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

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

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

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

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

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 $A$ ? What are the properties of $F[x]$ and/or $A$ 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