While I'm on the topic, here is another ring I need to show Euclidean. I'll show more of the work this time too. The ring is [itex]Z[{\sqrt 2 \over 2}(1+i)][/itex](adsbygoogle = window.adsbygoogle || []).push({});

So, using the standard norm difference approach, we pick any element alpha in the field and try and show we can always find an element beta in the ring of integers so that the norm of the difference is less than one. We can start like this:

[tex]

\alpha=a_1 + a_2 \sqrt 2 + a_3 i + a_4 i \sqrt 2

[/tex]

[tex]

\beta=b_1 + b_2 \sqrt 2 + b_3 i + b_4 {\sqrt 2 \over 2}(1+i)

[/tex]

[tex]

\gamma=\alpha-\beta

[/tex]

[tex]

\gamma=(a_1 - b_1) + (a_2 - b_2 - {1 \over 2}b_4) \sqrt 2 + (a_3 - b_3) i + (a_4 - {1 \over 2}b_4) i \sqrt 2

[/tex]

[tex]

\gamma=c_1 + c_2 \sqrt 2 + c_3 i + c_4 i \sqrt 2

[/tex]

After some pain and agony we get the norm of gamma:

[tex]

N(\gamma)=(c_1^2 - 2c_2^2 - c_3^2 + 2c_4^2)^2 + (2c_1c_3-4c_2c_4)^2

[/tex]

The best we can do is force c1, c2 and c3 to be less than 1/2 and c4 to be less than 1/4 (we can switch this for c2 and c4, but the norm is symmetric in c2 and c4, so this doesn't matter).

However, these values don't let you force the norm less than 1. I tried exapanding and recombining in different ways and still nothing.

Any clever ideas?

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# Another Euclidean ring

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