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I am given a polynomial $$ f = 2x^{2} + 4x + 6$$

and I have to determine in which of the rings, $\mathbb{Z}[x]$, $\mathbb{Q}[x]$, or $\mathbb{Z}_{5}[x]$, $f$ is a primitive polynomial, and I have to compute the content $C(f)$ in each case.

For $\mathbb{Z}[x]$, I was able to calculate $C(f)$ by using the Euclidean algorithm on the coefficients $2,4,6$, and found it to be $2$. Since $2$ is not a unit in $\mathbb{Z}$, I determined that $f$ is NOT primitive there.

For $\mathbb{Z}_{5}[x]$, I reduced all the coefficients $\mod 5$ and then found $C(f)$ to be equal to $1$, again, via the Euclidean algorithm. Since $1$ is a unit in $\mathbb{Z}_{5}$, I determined that $f$ is primitive there.

My question is what to do in the case of $\mathbb{Q}[x]$. Of course, all integers are also rationals, but can I use the same Euclidean algorithm to find the same $C(f)$, and then determine that because $2$ is a unit in $\mathbb{Q}$ that $f$ is primitive in $\mathbb{Q}[x]$? Or is there a different way to calculate the $\gcd$ of rational numbers?

Thanks.

  • https://math.stackexchange.com/q/151081 – Sahiba Arora Jun 01 '17 at 21:14
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    Hint: In a field every nonzero element is a unit, so every nonzero polynomial has a unit coefficient, and units have only unit divisors, so ... – Bill Dubuque Jun 01 '17 at 21:29
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    In a field every nonzero element is unit: this means that every nonzero polynomial is primitive. – Crostul Jun 01 '17 at 22:16
  • @Crostul Why spoil the hint? – Bill Dubuque Jun 02 '17 at 02:11
  • @SahibaArora so using that method, the $\gcd$ of $2$, $4$, and $6$ will be the same regardless of whether I consider them to be rationals or integers? –  Jun 21 '17 at 20:26
  • @BillDubuque I still need to calculate the content in $\mathbb{Q}[x]$ because I was asked to, regardless of how I determined whether the polynomial is primitive. Am I correct then that the content will be the same in $\mathbb{Q}[x]$ as it is in $\mathbb{Z}[x]$, since the $\gcd$ of the coefficients will be the same regardless of whether I treat them as rationals or integers? –  Jun 21 '17 at 20:29
  • @ALannister Absolutely. – Sahiba Arora Jun 21 '17 at 20:30
  • @ALannister If some gcd argument is a unit then the gcd divides that unit so it must also be a unit. Normally unit gcds are normalized to be $1$ (gcds are unique only up to associates, i.e. unit multiples). – Bill Dubuque Jun 21 '17 at 20:41

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