It is known that, under GRH, a real quadratic field is Euclidean iff it is a UFD. So, assuming the conjecture of Gauss and GRH, we expect that there are infinitely many Euclidean real quadratic fields.

I want to know if there are any approaches in showing that there are infinitely many Euclidean real quadratic fields by explicitly constructing the Euclidean functions. Can we hope for a single (or finitely many) Euclidean function(s) that will do the job or we need infinitely many different Euclidean functions? Can these Euclidean functions be obtained by modifying the norm function?

Since very few non-norm Euclidean quadratic fields are known, can we at all hope for showing the infinitude of real UFD quadratic fields by finding infinitely many Euclidean quadratic fields?

• Since we don't even know how to show that there are infinitely many real quadratic UFDs I cannot see how you can expect that you will get an answer to any of your questions. Oct 17, 2013 at 17:30
• By the way, we do not need GRH to conclude that a real quadratic number field with class number $1$ is Euclidean, except for at most two fields (this unconditional result is due to W. Narkiewicz). Oct 17, 2013 at 18:14
• @Franz Lemmermeyer:So showing eulideanity would be harder than showing uiqueness of factorization? I also thought that but wasn't so sure. So is this approach (showing euclideanity) of finding a large number of UFD's completely hopeless? Oct 18, 2013 at 12:23
• Well maybe there's something special an unexploited about Euclideanity that no one's thought of. But for UFDs you have discrete invariants like class numbers to play with and hundreds of years of wonderful analysis to work off of. Oct 18, 2013 at 13:32
• I'm not sure what you mean by "a single (or finitely many) Euclidean function(s)"; how do you tell whether two Euclidean functions for different number rings are the same or different? Anyway, this paper arxiv.org/abs/1106.0856 might interest you. Oct 21, 2013 at 18:58

Usually the euclidean property is proved indirectly, using sieve-theoretic methods (in particular, Wilson's large sieve over number fields and the Gupta-Murty lemma). For this kind of argument see Harper's work on $\mathbb{Q}(\sqrt{14})$ and Harper-Murty.
I think that the only known non-trivial example (that is, for fields which are not norm-Euclidean) of a real quadratic field with an Euclidean function is $\mathbb{Q}(\sqrt{69})$:
"Since very few non-norm Euclidean quadratic fields are known" I don't think this is true, depending on the meaning of "few" here. Malcolm Harper showed in his thesis that all real quadratic fields with class number $1$ and discriminant $\leq 500$ are Euclidean, but I believe this was never published.