Terminology. Can I say that $t^2 - 9$ is factorable by a "product of conjugates"?

Question

Can I say that $t^2 - 9$ is factorable by $(t -3)(t + 3)$, a "product of conjugates"? Is that correct terminology? It seems so ingrained in mathematical culture that in all of Wikipedia's definitions, there's only qualified conjugates such as "complex conjugate", but they all seem to conform what I'm inclined to say.

Answer 1

The answer to whether or not terminology is appropriate is very often; depends. If I had to give a direct answer, I would say no. Here is my reasoning.

In number theory, your terminology at least has a slight foothold. The reason is that in general, $t^2-n = (t-\sqrt{n})(t+\sqrt{n})$ for any $n\in\mathbb{Z}$, and the operation $a+b\sqrt{n}\mapsto a-b\sqrt{n}$, e.g. for $a,b\in\mathbb{Q}$, is usually called conjugation. It can be viewed as a generalization of complex conjugation for Gaussian integers. The problem is, for this to be a well-defined function, we need that $n$ is not a perfect square, which $9$ is. So in spirit, I can accept seeing this a form of conjugation, but in reality, it is not.

If you were factoring this polynomial while doing complex analysis, or indeed any setting outside the world of $\mathbb{Q}(\sqrt{n})$, I would stay away from this terminology, in fear of confusion with complex conjugation and the like.

You could define a new form of conjugation for polynomials, $(a_nt^n+\dots+a_1t+a_0)\mapsto (a_nt^n+\dots+a_1t-a_0)$. But in its essence, terminology is about communication, and if it's a terminology that might confuse people, it is bad terminology.