Is Gauss's lemma valid for polynomials with coefficients in a GCD domain?

Question

Wikipedia's proof of Gauss's lemma requires this theorem:

If $(C \mid S\cdot T) \land \lnot \operatorname{invertible}(C)$, $C$ has a non-invertible divisor in common with at least one of $S$ and $T$.

I can prove it for a Bézout domain, but not a GCD domain. Can anybody help me?

Original text:

If the contents $c = c(ST)$ is not invertible, it has a non-trivial divisor in common with the leading coefficient of at least one of $S$ and $T$ (since it divides their product, which is the leading coefficient of $ST$).

score 7 · Answer 1 · edited Jun 23 '22 at 18:52

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FYI: a proof of Gauss's Lemma for GCD domains is given in Section 15.5 of these notes. (I don't claim any superiority to any other proofs you may have seen or discovered...)

edited Jun 23 '22 at 18:52

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answered May 10 '11 at 20:00

Pete L. Clark

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That proof is actually identical to @Bill Dubuque's one. Now I see another proof based on $(ct,st)\mid(c,s)t \land (c,s)t\mid(ct,st)$… – beroal May 11 '11 at 17:04
@beroal Probably your "other" proof of the GCD distributive law is equivalent to the proof I gave above, except that you have torn it apart into two separate cases, one for each arrow direction. But that goes against the whole point of the proof - to unify both directions using the universal $\iff$ definition of the GCD, $\rm\ a\ |\ b,c\ \iff\ a\ |\ (b,c):.:$ If you master this technique, all these proofs become quite trivial. See the links I gave. – Bill Dubuque May 11 '11 at 17:20
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Note The proof in Pete's notes, excerpted from Haible: Gauss's Lemma without primes, 1990 is essentially the same as the OP's linked Wikipedia proof (attributed to Richman 1988). It's hinted exercise 1-6-8 in Kaplansky's Commutative Rings 1970, but is surely much older than that. It deserves to be better known that it can be proved more simply and more conceptually in the same way as the UFD case. I'll post this later when I have some spare time. – Bill Dubuque May 11 '11 at 18:17
@BillDubuque Sorry for asking about such an old post. May I ask that if you have written such a post. I'm truly looking forward to this simple and conceptual proof. Thanks in advance! – Wembley Inter Feb 27 '23 at 13:40

score 5 · Answer 2 · edited Apr 13 '17 at 12:19

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HINT $\ $ Simply apply Euclid's Lemma, i.e. $\rm\ (c,s)= 1,\ c\ |\ st\ \Rightarrow\ c\ |\ (ct,st) = (c,s)\:t = t\:.$

Alternatively if $\rm\ (c,s) = 1 = (c,t)\:$ then $\rm\ c\:|\:st\ \Rightarrow\ c\ |\ (cc,cs,ct,st) = (c,s)(c,t) = 1\:.$

Per request, here's a simple proof of the GCD distributive law $\rm\ (a,b)\:c = (ac,bc)\:.$

LEMMA $\rm\ \ (a,b)\ =\ (ac,bc)/c\quad$ if $\rm\ (ac,bc)\ $ exists $\rm\quad$ (GCD distributive law )

Proof $\rm\quad d\ |\ a,b\ \iff\ dc\ |\ ac,bc\ \iff\ dc\ |\ (ac,bc)\ \iff\ d|(ac,bc)/c$

The above proof uses the universal definitions of GCD, LCM, which often serve to simplify proofs, e.g. see a similar proof of the GCD * LCM law..

edited Apr 13 '17 at 12:19

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answered May 09 '11 at 20:53

Bill Dubuque

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Can you please prove $(ct,st)=(c,s)t$? I see why $(c,s)t$ is a common divisor of $ct,st$, but why it is the greatest common divisor? – beroal May 10 '11 at 13:09
@beroal: this follows directly from the definition of 'ideal generated by', as $xct+yst=(xc+ys)t$. – wildildildlife May 10 '11 at 16:51
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@wil No, GCD domains generally are not Bezout, i,e. finitely generated ideals needn't be principal. – Bill Dubuque May 10 '11 at 18:00
It turned out quite simple. Thanks. – beroal May 11 '11 at 17:05

Is Gauss's lemma valid for polynomials with coefficients in a GCD domain?

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