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I'm trying to construct a polynomial $f(x)$ such that $f(x)$ is reducible over $\mathbb{Z}$ and $\mathbb{R}$ but irreducible in $\mathbb{Q}$.

So far I've constructed a polynomial $f(x) = x^2 + 2x + 2$, which I just want to know how to check if it is reducible or irreducible over $\mathbb{R}$ and $\mathbb{Z}$. Also I would like some advice on how to construct a polynomial that would satisfy the conditions given if such a polynomial exists.

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  • $\begingroup$ An irreducible polynomial over $\mathbb{R}$ is either of degree $1$ or $2$. $\endgroup$ – Clément Guérin Feb 19 '16 at 8:13
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    $\begingroup$ Your polynomial is reducible over $\mathbb{Q}$ (you constructed it using a factorisation over $\mathbb{Q}$). $\endgroup$ – Clément Guérin Feb 19 '16 at 8:14
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    $\begingroup$ Note that any polynomial of the form $2p(x)$ is reducible over $\Bbb Z$ but not necessarily $\Bbb Q$. $\endgroup$ – Arthur Feb 19 '16 at 8:14
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    $\begingroup$ @Jmath99, try $4X^3-8$ ? $\endgroup$ – Clément Guérin Feb 19 '16 at 8:16
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    $\begingroup$ As per Gauss' Lemma and friends irreducibility of a polynomial over $\Bbb{Z}$ is equivalent to irreducibility over $\Bbb{Q}$ except for eventual constant factors (those may be irreducible in $\Bbb{Z}[x]$ but are necessarily units in $\Bbb{Q}[x]$). $\endgroup$ – Jyrki Lahtonen Feb 19 '16 at 8:42
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I wonder if there is anything more basic than $2 x^2 - 4$.

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Nizar Feb 19 '16 at 9:36
  • $\begingroup$ @Nizar: Oh, I should have said: "What is $2 x^2 - 4$ ? " $\endgroup$ – Orest Bucicovschi Feb 19 '16 at 10:13
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I think I found a solution,$f(x) = 4x^3 + 12x^2 + 12x + 12$. Could someone verify if this polynomial satisfies the given conditions? I know $f(x)$ is cubic so is reducible over $\mathbb{R}$, and irreducible in $\mathbb{Q}$ by Eisenstein's Criteria.

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