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I don't understand this sentence.

  • Give an example of polynomial in $\Bbb Z [x]$ that is irreducible in $\Bbb Q [x]$ but factors when reduced mod $2,3,4$ and $5$

What does "reduced mod $2,3,4$ and $5$" means?

Watson
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이정태
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  • For instance $2X^2+3X+7$, when reduced mod $3$, becomes $\bar 2 X^2 + \bar 1$, where $\bar a$ denotes $a$ mod $3$. – Watson Dec 01 '16 at 12:49
  • For example: the polynomial $x^2-4x-1$ is irreducible in $\mathbb Q[x]$ (as the real roots are not rational) but $\pmod 2$ it is $(x-1)^2$. – lulu Dec 01 '16 at 12:49
  • Related (different since you are asking also mod $4$): https://math.stackexchange.com/questions/77155/ – Watson Dec 01 '16 at 12:52
  • Then "reduced mod 2,3,4 and 5" means that polynomial f (x) is reducible in Z_2 [x], Z_3 [x], Z_4 [x], and Z_5 [x] ? – 이정태 Dec 01 '16 at 12:52
  • @이정태 : "factors" means "is reducible", or "is not irreducible", but "reduced mod $n$" means that you consider the coefficients in $\Bbb Z/n\Bbb Z$. These notions are not related: $x^2+x+1$, when reduced mod $2$, is irreducible. – Watson Dec 01 '16 at 12:54

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A polynomial is reducible if you can factor it. Otherwise it's irreducible. Notice that $x^2+1$ is irreducible over $\mathbb{Q}$, but $(x+1)(x+1) \equiv x^2+2x+1 \equiv x^2+1 (\bmod{2})$, so $x^2+1$ is reducible over $\mathbb{Z}_2$.

B. Goddard
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