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In my class notes my teacher defined a polynomial with coefficients in a ring $R$ as:

If $R$ is a ring a polynomial with coefficients in a ring $R$ is a formal expression: $$a_o + a_1 x + ... + a_n x^n$$ with $n \in \mathbb N_0$ and $a_i \in R$

I also checked the definition given in some Abstract algebra books and all of them defined it as a formal expression. Why is that so? Isn't just enough to say that a polynomial if a function:$$f:R\to R$$ $$x \mapsto a_0 + a_1x+...+a_nx^n$$

Why define a polynomial as a formal expression?

Eduardo Magalhães
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    Because the polynomials $f(x) = 0$ and $g(x) = x+x^2$ should be distinct elements of $\mathbb{Z}_2[x]$. – Randall Jan 11 '22 at 17:12
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    Two different polynomials can define the same function. For example if the field is $\Bbb F_2$, then the functions $y=x$ and $y=x^2$ are the same. – markvs Jan 11 '22 at 17:12
  • There is no concept of evaluation of a polynomial in a polynomial ring. The $x^i$s are not variables to which you can assign a value. – John Douma Jan 11 '22 at 17:22
  • @JohnDouma There most certainly is a concept of evaluation of (formal) polynomials - and it is fundamental. – Bill Dubuque Jan 11 '22 at 19:58

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Because these definitions are not equivalent. For example, consider the field $\mathbb{F_2}$. The polynomials $x+1$ and $x^2+1$ are equal as functions. But they are not the same polynomial. And we don't want them to be the same polynomial.

Mark
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    This would be improved by any hint of why we don’t want them to be the same polynomial. For instance, polynomial functions can’t generally be evaluated in larger rings or fields, but that’s something we constantly want to do, much like plugging a complex number into a real polynomial. – Kevin Carlson Jan 11 '22 at 17:28
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    Please strive not to add more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. – Bill Dubuque Jan 11 '22 at 19:31