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,
$$2x^3 – 10$$
$$2(x^3 – 5)$$
$$…$$
Best Answer
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.