Factor irreducible polynomial in Z[x] and R[x]

Question

I've got a couple of problems from an old exam in abstract algebra that I have difficulty in understanding.

1) Write the polynomial $2x^3 - 10$ as a product of irreducible elements in $\mathbb{Z}[x]$, and list the irreducible elements in this factorization.

2) Write the polynomial $2x^3 - 10$ as a product of irreducible polynomials in $\mathbb{Q}[x]$, and list the irreducible polynomials in this factorization.

Should I set $F_1 = \mathbb{Z}[x]$ and $F_2 = \mathbb{Q}[x]$ and do polynomial division to find it with the extension fields, and list the elements in the field? Even so, I don't see how this can be done for the $\mathbb{Z}[x]$.

Obviously, $$2x^3 - 10$$ $$2(x^3 - 5)$$ $$...$$

Answer 1

Over $\mathbb{Z}[x]$, both $2$ and $x^3-5$ are irreducible, so you can't factor further than $$ 2x^3 - 10 = 2(x^3 - 5) \text{.} $$ The reason you can't factor $x^3 - 5$ further is that if it had any more factors, one of them would need to have degree $1$, meaning $x^3 - 5$ would have a zero in $\mathbb{Z}$, which it doesn't.

The same argument works over $\mathbb{Q}[x]$, but you now need to use that $\sqrt[3]{5}$ isn't rational.

Answer 2

First of all, we notice that the polynomial does not have integer or rational roots (since thos would be one of -5, 5, -1, 1).

The polynomial is not irreducible in $Z[x]$. From the definition of irreducible polynomials: for $f(x), a(x), b(x)$ that belong to $R[x]$, $R$ ring, if $f(x)$ is not invertible, it is called irreducible if

$f(x)=a(x)b(x) => a(x)$ invertible or $b(x)$ invertible.

So in $Z[x]$, since neither $2$ nor $x^3-5$ is invertible, the polynomial is not irreducible.

From the same definition, it is irreducible in $Q[x]$, since 2 is invertible in $Q[x]$.