Factorizing polynomials to irreducibility only

Question

Consider e.g. $2x^2-1 \in \mathbb{Q}[x]$.

It appears to meet the definition of irreducible:

A non-constant polynomial is irreducible over $\mathbb{Q}$ if its coefficients belong to $\mathbb{Q}$ and it cannot be factored into the product of two non-constant polynomials with coefficients in $\mathbb{Q}$.

However, when passing this to a CAS factorize method (making sure to specify that it should be done in $\mathbb{Q}[x]$), it is factorized to: $$2 \left(x^2-\frac{1}{2}\right),$$ which is a product of a constant and a non-constant.

So which factorization form is 'correct'? When is the factorization algorithm supposed to terminate? I would prefer to factorize on $\mathbb{Q}[x]$ down to $2x^2-1$, rather than this more 'final' form. Is there a consistent name for this form of factorization?

Answer 1

Factorization can only ever be unique up to a "unit", which is an invertible element of the ring. In the case of the polynomial ring $\mathbb{Q}[x]$, the invertible elements are polynomials of degree zero (i.e. rational numbers except $0$). So:

1. Yes, that polynomial is indeed irreducible
2. typically, computer algebra systems will prefer normalized polynomials (i.e. the highest coefficient equal to $1$)
3. If you are interested in factorization over whole numbers (i.e. $\mathbb{Z}[x]$ instead of $\mathbb{Q}[x]$), the situation changes dramatically changes, then the only freedom is to put a $(-1)$ in front (because $\pm1$ are the only two units in $\mathbb{Z}$)

Note: the same kind of ambiguity happens already when factoring whole numbers into primes: 6=(2)(3)=(-2)(-3) and such. The convention "always use positive numbers" is analog to "always use normalized polynomials")