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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?

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    $\begingroup$ I think the gcd is usually defined as the monic polynomial of highest degree that divides both. $\endgroup$
    – Steve Kass
    Commented Jul 3, 2020 at 21:12

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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 $\mathbf Z[x]$.

The idea of being relatively prime (not saying what a general gcd is!) is that the only common factors are units. In a UFD we say two elements are relatively prime when their only common factors are units and we may abbreviate this to "gcd = 1" but everyone is expected to realize the concept is only defined up to scaling by a unit. There are other settings where there is unique factorization of ideals, not of elements, so you have to understand what kind of gcd is intended when you see that term used.

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.

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