2
$\begingroup$

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?

  • 0
    "Can we have an **irreducible** polynomial over an integral domain having no roots in the domain?" do you mean **reducible** ?2012-10-21
  • 0
    Yes, I meant reducible.2012-10-21

1 Answers 1

3

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

  • 0
    Thanks. But shouldnot factorization be in lower degree than 2?2012-10-21
  • 1
    @Reader - why ? I recall that the definiton only requires that both polynimials are $\neq 0$ and that they are not invertible2012-10-21
  • 0
    A 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
  • 0
    @Reader - What is $F$ ?2012-10-21
  • 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
  • 0
    @AndréNicolas - nice example!2012-10-21
  • 0
    Thanks. 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
  • 0
    ֲ@Reader en.wikipedia.org/wiki/Irreducible_polynomial#Generalization2012-10-21
  • 0
    Thanks Belgi and André Nicolas2012-10-21
  • 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
  • 0
    there is no v,I would like to do it if I can find it.2012-10-21
  • 0
    I think I have clicked somewhere now, have I done it or not?2012-10-21
  • 0
    Can somebody tell me what does it means to click on V? I am new at this forum.2012-10-21
  • 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
  • 0
    @ Belgi Oh I see.2012-10-21