Why are integer GCDs positive? [unit normalization of GCDs]

Question

The definition in my text reads,

An integer $d$ is said to be the greatest common divisor of two non-zero integers $a$ and $b$ iff,

$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $k|d$

Now here's the thing, if $d|a$ and $d|b$ then surely $-d|a$ and $-d|b$ as well, also $k|-d$

What I take from this? GCD is not unique! That is if $\mathrm{gcd}(12,8)= 4$ then by the definition, $\mathrm{gcd}(12,8) = -4$ as well.

Yet I never ever seen a negative gcd. Someone please explain.

Maybe, $4>-4$, and we want the "greatest common factor" so...? But that still doesn't justify the definition.

Answer 1

You are right, it is all about definitions. With the definition you gave the gcd is really not unique and it might be negative as well. But because most of the time we are using only the positive gcd then some simply prefer to add the words "$d$ is positive" to your definition or just give other definitions. For example a very common definition of gcd in number theory is "$d|a$ and $d|b$ and if $k$ is any other common divisor of $a$ and $b$ then $k\leq d$". So that definition already requires the gcd to be positive.

Answer 2

The text is employing the $\color{#c00}{\rm universal}$ definition of a gcd, namely

$$\ c\mid a,b \!\!\color{#c00}{\overset{\rm u\!\!}\iff}\! c\mid \gcd(a,b)\qquad$$

Direction $(\Leftarrow)$ implies a gcd is a common divisor of $a,b,\,$ by choosing $ c = \gcd(a,b),\,$ and the reverse direction $(\Rightarrow)$ implies that a gcd is "greatest" w.r.t. divisibility order, i.e. it's divisible by all other common divisors $\,c\,$ of $a,b\,$ (so has greatest magnitude in $\Bbb Z,$ and greatest degree in $K[x])$.

Generally a gcd is not unique: if $\,d,d'$ are both gcds of $\,a,b\,$ then $\, c\mid d\!\!\!\color{#c00}{\overset{\rm u\!\!}\iff}\! c\mid a,b\!\!\!\color{#c00}{\overset{\rm u\!\!}\iff}\! c\mid d'\,$ so specializing $\,c =d\,$ and $\,c = d'\,$ shows $\,d\mid d'\mid d,\,$ i.e. $\,d\sim d'\,$ are associate (divide each other). And conversely too: $\,d=\gcd(a,b)\,$ is associate to $\, d'\,$ then $\,d\mid d'\mid d,\,$ so $\,c\mid d\!\iff\! c\mid d',\,$ thus $\,d'$ is also a gcd of $\,a,b.\,$ In an integral domain $\,a\,$ is associate to $\,b\!\iff\!$ $\,a = ub\,$ where $\,u\,$ is a unit (= invertible), i.e. associates are unit multiples. Thus gcds are unique up to unit multiples.

In some rings with simple unit group structure we can choose canonical representatives of associate classes, which allows is to choose normal forms for gcds, e.g. in $\,\Bbb Z\,$ (with units $\pm 1)$ we normalize gcds $\ge 0,\,$ and in a polynomial ring $\,K[x]\,$ over a field (units = constants $0\neq c\in K) $ we normalize polynomial gcds to be monic (lead coeff $\,c_n = 1),\,$ by scaling the polynomial by $\,c_n^{-1}\,$ if need be (so a constant gcd $\,c_0\neq 0\,$ normalizes to $1).\,$ Hence in both cases we can say that two elements are coprime $\!\iff\!$ their gcd $= 1$ (vs. a unit). Such normalizations are sometimes called unit normal representatives in the literature.

Note on notation (abuse of language). Some writers (implicitly) use the convention that $\gcd(a,b) = c\,$ means $\,\gcd(a,b)\approx c,\,$ i.e. the gcd is associate to $c,\,$ even when there is no natural choice of unit normalization. This is similar to other such abuses of language that - for convenience - blur the distinction between an equivalence class and a (non-canonical) rep of the class. Common simple examples are fractions: $\,x = a/b,\,$ meaning $\,x\approx a/b\,$ or $\, x = [a/b],\,$ and elements of quotient rings, e.g. in $\,R = \Bbb Z/n\,$ one often denotes the coset $[k] = k+n\Bbb Z\,$ by $\,k\,$ and $\,j = k\,$ means $\,[j] = [k]\,$ (or $\,j\equiv_n k\,$ when using congruences), and such notational abuse is often (naturally) extended to $R$-algebras over such quotient rings, e.g. polynomial and power series rings, where we often see $\,ax^2+bx+c\,$ vs. $[a] x^2 + [b]x + [c],\,$ and similarly for product and matrix rings (tuple notation), etc. Without such, notational complexity might serve to obfuscate the key idea(s) under discussion.