Let's suppose that every element of

is reducible.

That means whenever

and

and

are not units.

That would mean that every degree 2 polynomial could be factored into two linear polynomial's with coeffeints in the reals.

i.e

But we know from the quadratic formula that is

that the polynomial can only be factored if we use coeffeints from the complex numbers.

So if

then the only factorization over the real numbers would be of the form.

Since

n is a unit in the reals(every non zero real number is a unit)

We can only factor it as a unit multiplied by some element of