# Definition of relatively prime polynomials over $\mathbb{Q}[x]$

So I see that various sources online define two polynomials $$f,g\in\mathbb{Q}[x]$$ to be relatively prime provided $$\gcd(f,g)=1$$. However, isn't it the case that the greatest common divisor for polynomials over a field is unique up to multiplication by a non-zero constant? My question is, shouldn't we define $$f,g\in\mathbb{Q}[x]$$ to be relatively prime provided their GCD is a unit? I was computing some examples with the Euclidean Algorithm by hand, and for example, in one case I obtained a GCD of $$3/4$$, but Sage and WolframAlpha say the GCD is $$1$$. Is it just typical convention to take the GCD to be $$1$$ in these cases? I mean, it is obvious that in $$\mathbb{Q}[x]$$, any non-zero rational can be a "common divisor" of two polynomials. So I suppose the notion of "greatness" here for GCD is the degree. With all of this in mind:

What is the most standard way to define relatively prime polynomials over $$\mathbb{Q}[x]$$? What about arbitrary polynomial rings? Maybe define relatively prime if the ideal they generate is the whole ring?

• I think the gcd is usually defined as the monic polynomial of highest degree that divides both. Commented Jul 3, 2020 at 21:12

The notion of gcd is defined up to multiplication by a unit. In particular, when the only common factors are units we may say "gcd = 1" but everyone is expected to realize that means "gcd is a unit". To make a definite choice of gcd for polynomials in $$F[x]$$ where $$F$$ is a field, it is conventional to regard "the" gcd as being the monic one, but as you already observed, the last nonzero remainder in Euclid's algorithm in $$F[x]$$ need not be monic. So you just rescale it to be monic. Math has many abuses of notation and terminology. The terminology "gcd = $$1$$" is one of them. You accept what it really means and move on with learning math.
There is no single all-encompassing notion of gcd. The ring $$\mathbf Z[x]$$ is a UFD but it is not a PID, so you can't say "relatively prime" should mean the elements generate the whole ring. For example, $$2$$ and $$x$$ are relatively prime nonunits in $$\mathbf Z[x]$$ but the ideal $$(2,x)$$ is a proper ideal. You can't write $$1 = 2u(x) + xv(x)$$ for some $$u(x)$$ and $$v(x)$$ in $$\mathbf Z[x]$$.
In a Dedekind domain, two ideals $$\mathfrak a$$ and $$\mathfrak b$$ are relatively prime when their only common ideal factor is the unit ideal $$(1)$$, which is equivalent to $$\mathfrak a + \mathfrak b = (1)$$. For instance, in $$\mathbf Z[\sqrt{-5}]$$ the elements $$2$$ and $$1+\sqrt{-5}$$ have no common factors besides $$\pm 1$$ but the ideals $$(2)$$ and $$(1+\sqrt{-5})$$ have a nontrivial common ideal factor, namely the prime ideal $$(2,1+\sqrt{-5})$$. Thus $$2$$ and $$1+\sqrt{-5}$$ are relatively prime elements of $$\mathbf Z[\sqrt{-5}]$$ but $$(2)$$ and $$(1+\sqrt{-5})$$ are not relatively prime ideals.