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I know this is probably a dumb question. Since a field is a PID then it should be a GCD domain according to the chain of domains. However, take $\mathbb{Q}$ for an example, I don't think it is a GCD because I can't find the greatest common divisor of $5$ and $7$. I mean, every nonzero element divides $5$ and $7$ since it is a field, and I can't find the largest one.

Coco
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  • Is it that take $a$ and $b$ which are not both zero, every non-zero element is a GCD of $a$ and $b$? – Coco Apr 04 '23 at 04:49
  • In generic domains, GCDs are never meant to be unique (or canonically determined). They are greatest in the sense that they are $d$-s such that, for all $y$ such that $y\mid a$ and $y\mid b$, $y\mid d$. Note that $\mid$ is not an order relation. – Sassatelli Giulio Apr 04 '23 at 04:53
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    The "greatest common divisor" is "the greatest" in terms of divisibility, i.e. every other common divisor divides it. This has nothing to do with the size of the number. The GCDs are not uniquely determined, but only up to equivalence (two elements are equivalent if they divide each other). Thus, in $\mathbb Z$, the GCD's of $5$ and $7$ are $1$ and $-1$; in $\mathbb Q$, the GCDs of $5$ and $7$ are all nonzero rational numbers, and they are all equivalent to each other. –  Apr 04 '23 at 04:54
  • The gcd $d$ of any set $S$ containing a unit $u$ (invertible) is $1$, since $d\mid u\Rightarrow d\mid uu^{-1}=1,,$ so $,d,$ is a unit (and unit gcds are normalized to $1)$. In a field every element $\neq 0$ is a unit, so every set $S\neq {0}$ contains a unit, so $\gcd S = 1.\ \ $ – Bill Dubuque Apr 04 '23 at 06:50
  • @StinkingBishop Thank you!!! – Coco Apr 04 '23 at 20:02

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