962: For Polynomials Ring over Integral Domain, Unit Is Nonzero Constant

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description/proof of that for polynomials ring over integral domain, unit is nonzero constant

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Topics

About: ring

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note
• 3: Proof

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Starting Context

• The reader knows a definition of polynomials ring over commutative ring.
• The reader knows a definition of integral domain.
• The reader knows a definition of units of ring.

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Target Context

• The reader will have a description and a proof of the proposition that for the polynomials ring over any integral domain, any unit is a nonzero constant.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$R$: $\in \{\text{ the integral rings }\}$
$R[x]$: $=\text{ the polynomials ring over }R$
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Statements:
$\{\text{ the units of }R[x]\}\subseteq \{\text{ the nonzero constants in }R[x]\}$
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2: Note

As $R$ is not necessarily a field, a constant may not be any unit: for a $p(x)=p_0\in R[x]$, $p_0\in R$ may not have any inverse, then, $p(x)=p_0\in R[x]$ will not have any inverse: compare with the proposition that for the polynomials ring over any field, the units are the nonzero constants.

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3: Proof

Whole Strategy: Step 1: see that each unit is a nonzero constant.

Step 1:

$0\in R[x]$ is not any unit.

Let $p(x)=p_n{x}^n+...+p_0\in R[x]$, where $p_n\ne 0$, be any unit.

There is an inverse, $p^{\prime }(x)=p_m^{\prime }{x}^n+...+p_0^{\prime }\in R[x]$, where $p_m^{\prime }\ne 0$, such that $p(x)p^{\prime }(x)=p^{\prime }(x)p(x)=1$.

$p(x)p^{\prime }(x)$ has the $p_n{p}_m^{\prime }{x}^{n+m}$ term, but $p_n{p}_m^{\prime }\ne 0$, because $R$ is an integral domain.

So, $n+m=0$, because $p(x)p^{\prime }(x)=1$. That means that $n=0$ and $m=0$, which means that $p(x)$ is a nonzero constant.

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References

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