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How are these two sentences equivalent?

  1. Let $f(x)\in\mathbb{Z}[x]$ be a polynomial such that it is the multiplication of two polynomials with rational coefficients. Then $f(x)$ is also the multiplication of two polynomials with integer coefficients.
  2. If $f(x)$ is irreducible on $\mathbb{Q}$, then $f(x)$ is irreducible on $\mathbb{Z}$.
Mill
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  • Well, the second statement in completely trivial as $;\Bbb Z\subset \Bbb Q;$ . Perhaps you want the other direction, with the added condition that $;f;$ is an integer polynomial...? – DonAntonio Jun 01 '14 at 14:02
  • This statement is known as Gauss's Lemma, you can find your answers by starting here: http://en.wikipedia.org/wiki/Gauss's_lemma_(polynomial) – o0BlueBeast0o Jun 01 '14 at 14:06
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    The first statement is also trivial because $f=f\cdot 1$. – hmakholm left over Monica Jun 01 '14 at 14:06
  • I don't want proof to neither. I just want to know how statements 1 and 2 are equivalent? – Mill Jun 01 '14 at 14:09
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    The polynomial $2x$ is irreducible in $\mathbb{Q}[x]$, but not in $\mathbb{Z}[x]$. You need some more hypotheses. – egreg Jun 01 '14 at 14:10

1 Answers1

1

Both are true statements, so they are equivalent.

In general let $D$ be an integral domain, then $f\in D[x]$ is irreducible iff $f\in F[x]$ is irreducible where $F$ is the field of quotient of $D$. This can be proved using Gauss Lemma.

mez
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