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Given a ring $R$, the polynomial ring is defined as

$$R [x]:=\left\{\sum_{k=0}^n a_k x^k: n\ge0, \ a_k \in R \ \text{for }k\in \left\{0, 1, \dots, n \right\}\right\}. $$ However, it is not usually specified what $x$ is. In order for multiplication to make sense, I guess it has to be an element in $R$ at least. But is $R[x]$ the set of all functions $ P: R\rightarrow R, \text{given by } x\mapsto \sum_{k=0}^n a_k x^k$, or the set of those functions evaluated at $x$?

Sometimes $R$ is required to be commutative. Does that make any difference for R[x]?

mathslover
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  • See the linked dupes for rigorous constructions of polynomial rings, and the difference between formal vs. functional polynomials. Please search for answers before posing questions. – Bill Dubuque Feb 09 '22 at 09:53

2 Answers2

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We think of polynomials as infinite lists which have a finite point in after which everything is $0$. (Meaning that a polynomial is of finite degree by definition.)

For example the polynomial $x^2+1$ is actually the infinite list $(1,0,1,0,0,...)$. (Think of the first element as the $0$th element, then $n$th element in the list corresponds to the coefficient of $x^n$, notice that the list can contain numbers or more generally, elements of any ring.)

The operations are defined on this concept but this writing is not practical so we switch to the notation with $x$.

Usually the books just say $x$ to be an indeterminate.

Ata
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  • Thanks. So the confusion becomes from that polynomials are not the same as polynomial functions? Would it be accurate to say that $(a_0, a_1, ..., a_n, 0, ...)$ is a polynomial (where $0, a_0, a_1, ..., a_n$ are elements in R), and the map $P: R \rightarrow R$, given by $x \mapsto (a_0, a_1, ..., a_n, 0, ...) (1, x, ..., x^n, 0, ...)^T$, is a polynomial function? So can te list be interpreted as infinite row vectors? – mathslover Feb 09 '22 at 11:08
  • Or maybe $x \mapsto \left(a_0, a_1, \dots, a_n, 0, \dots\right) \left(1, x, \dots, x^n, x^{n+1}, \dots \right)^T $ – mathslover Feb 09 '22 at 11:19
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    I don't remember the usual-formal definition for a polynomial function but what you offer seems to be a great one! – Ata Feb 09 '22 at 11:21
  • So I could think of polynomials as elements of the set $ R \times R \times R \times \dots$ and denote this set by $ R[x] $? – mathslover Feb 09 '22 at 11:28
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    Yes that would be another way of reading the definition. I suggest you to search for how the formal definition is done in a formal text. – Ata Feb 09 '22 at 11:46
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Some hints:

First, $x$ is an unknown over $R$. This means that $x$ does not satisfy any nontrivial equation over $R$: $\sum_{k=0}^n a_kx^k=0$ with not all $a_k$ vanishing.

Second, consider the set of all functions $P:x\mapsto \sum_{k=0}^n a_kx^k$. If $R$ is an infinite field of char 0, two different polynomials define different polynomial functions, but this property is false for finite fields.

Third, if $R$ is commutative, then $R[x]$ is commutative.

Wuestenfux
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