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I am learning Irreducibility of polynomials, and I am reading this entry of Wiki. However, I am confused why the polynomial $p_{3}(x) = 9x^{2}-3=3(x\sqrt{3}-1)(x\sqrt{3}+1)$ is reducible over integers. I though it falls right into the defintion of irreducibility(over integer), because it could only be written into two polynomials of strictly lower degree with coefficients not being integers. The entry explains that it is reducible because the factor 3 is not invertible in integers, but I just can't see how this makes the polynomial reducible. I am very thankful if someone could help explain it.

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  • $\begingroup$ Recall when is a polynomial in $\mathbb Z[X]$ reducible in $\mathbb Z [X]$ $\endgroup$ – Soumik Mar 10 '19 at 7:54
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Well, the polynomial $f(x)=3(x^2-1)$ is reducible over the integers, since it factors into a product of non-units in ${\Bbb Z}[x]$. The units in ${\Bbb Z}[x]$ coincide with the units in ${\Bbb Z}$ and are $\pm 1$.

The definition says that a polynomial $f\in R[x]$ is irreducible if it can only be factored as $f=gh$ with $g,h\in R[x]$ such that one of $g,h$ is a unit in $R[x]$. The units in $R[X]$ are the units in $R$.

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    $\begingroup$ It is a confusing entry though, since the eye leaps to the factorisation into linear factors! $\endgroup$ – ancient mathematician Mar 10 '19 at 7:57
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    $\begingroup$ Indeed, the wiki entry is confusing. It says that if F is a field, a non-constant polynomial is irreducible over F if its coefficients belong to F and it cannot be factored into the product of two NON-CONSTANT polynomials with coefficients in F. Over a ring R, not each constant polynomial is a unit. – $\endgroup$ – Wuestenfux Mar 10 '19 at 8:33
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    $\begingroup$ Your argument is correct, but you should have applied it to $p_3(x) = 3(3x^2-1)$. $\endgroup$ – Paul Frost Mar 10 '19 at 9:03
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The polynomial p3 (x) is irreducible polynomial over integers but not irreducible element in the ring Z [x] as it can be written as 3 (x^2-1) where neither 3 is unit nor (x^2-1) is unit as definition of irreducible element is non- zero, non- unit element which whenever can be represented as product of two elements of the ring one of the element in that product must be unit. In wiki , the definition seems unclear then.

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