It is well-known that a degree 2 or 3 polynomial over a field is reducible if and only if it has a root. But what about integral domains? Can we have a reducible polynomial over an integral domain having no roots in the domain?
irreducible polynomial of degree 2 or 3 without roots in an integral domain.
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polynomials
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0"Can we have an **irreducible** polynomial over an integral domain having no roots in the domain?" do you mean **reducible** ? – 2012-10-21
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0Yes, I meant reducible. – 2012-10-21
1 Answers
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For example take $\mathbb{Z}$ and the polynomial $3(x^{2}+1)\in\mathbb{Z}[x]$
This polynomial have no roots in $\mathbb{Z}$ but $3\cdot(x^{2}+1)$ is a factorization
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0Thanks. But shouldnot factorization be in lower degree than 2? – 2012-10-21
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1@Reader - why ? I recall that the definiton only requires that both polynimials are $\neq 0$ and that they are not invertible – 2012-10-21
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0A polynomial $f (x) \in F [x]$ is called irreducible over $F$ if $deg(f) > 0$ and if its only factors are $c$ and $cf(x)$, where $c \in F$, $c\neq 0$, is any non-zero constant. – 2012-10-21
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0@Reader - What is $F$ ? – 2012-10-21
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2@Reader: If you don't like the example, look at $4x^2-4x+1=(2x-1)(2x-1)$. Certainly reducible over $\mathbb{Z}$, but no integer zeros. – 2012-10-21
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0@AndréNicolas - nice example! – 2012-10-21
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0Thanks. F is a field. What is definition of irreducible polynomial if F is not a field like, ring integral domain so on. – 2012-10-21
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0ֲ@Reader en.wikipedia.org/wiki/Irreducible_polynomial#Generalization – 2012-10-21
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0Thanks Belgi and André Nicolas – 2012-10-21
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0@Reader If you have found this answer helpful, I'd appriciate it if you could please accept by pressing on the v sign near the number 1 (left from my answer) – 2012-10-21
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0there is no v,I would like to do it if I can find it. – 2012-10-21
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0I think I have clicked somewhere now, have I done it or not? – 2012-10-21
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0Can somebody tell me what does it means to click on V? I am new at this forum. – 2012-10-21
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0@Reader - A. you can accept one answer (but you can un-accpt mine and choose anothr if you want) B. André posted a **coment** not an answer so you can not accept it unless he posts an answer (which I think he should!) – 2012-10-21
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0@ Belgi Oh I see. – 2012-10-21