Infinite integral domain with finite characteristic.

Asked Aug 21 '19 at 16:49

Active Aug 21 '19 at 17:40

Viewed 634 times

$\mathbb{F}_p(X)=\Big \{\frac {f}{g} \:\Big|\: f,g \in \mathbb{F}_p[X], g\neq 0 \Big \}$

Where $p$ is prime, it is an infinite field with finite characteristic.

I need an example of an infinite integral domain which is not a field with finite characteristic.

edited Aug 21 '19 at 17:40

Bernard

asked Aug 21 '19 at 16:49

THIRUMAL 5688

2

$\mathbb{F}_p[X]$? – Aphelli Aug 21 '19 at 16:51
Rational functions coefficients in $\mathbb{F}_p$ – THIRUMAL 5688 Aug 21 '19 at 16:56
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@ThirumalPachaiyappan I think that Mindlack provided the example you're looking for... namely $\mathbb{F}_p[X]$. – mathcounterexamples.net Aug 21 '19 at 17:04
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No, I mean that $\mathbb{F}_p[X]$ is an infinite integral domain with finite characteristic but is not a field. – Aphelli Aug 21 '19 at 17:04
As said, just the polynomial ring over a finite field. – Wuestenfux Aug 21 '19 at 17:52
Given the possibility of coefficients cancelling, it perhaps takes a smidgen of work to confirm that $\Bbb F_p[X]$ is in fact an integral domain (consider the lowest coefficients of any product) and is not a field (consider the highest coefficients of any product). – Robert Shore Aug 21 '19 at 18:48
Next time please search for similar questions before posting. – rschwieb Aug 21 '19 at 21:08

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