Proof that Z[(1+sqrt(-19))/2] is not a Euclidean domain

[@juliefinkjulesheartmagic1111]

Thank you for the awesome video! You’re so calm and clear.

[@bjorncarlsson787]

Slightly over my head but fascinating! Quadratic integers are intruiging. One would get the same result for Z[(1+√-43)/2]? And for some reason Z[√-13] is neither eucludian or a UFD.

[@coconutmath4928]

Thanks! They are an interesting and still active topic. The first example you gave should follow from a similar argument. In particular, the ideals and are still maximal ideals in that ring (you can check this using Legendre symbols for any prime p).
The second example is harder because isn't prime in that ring.

[@bjorncarlsson787]

@@coconutmath4928 you mean like e.g. 2 divides
(1+3√-13)(1-3√-13) but none of these numbers, which are both irreducible. And 2 is also irreducible.