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Proving or Disproving Claims

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Please see the attached file for the complete problem description.

Note: a) and b) are independent; for each of them it should be separately decided if that combination is an ideal and for the first point of the question all ideals are in Z[sqrt(-5)].

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Solution Summary

In this solution, it is shown how to prove for disprove the given claims.

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Please see the attached file for the complete solution response.

I'm attaching the proofs in .docx and .pdf format.

Prove or disprove the following claims: (please see the attached file)
The lattice of integer combinations of the given vectors is an ideal: (please see the attached file)
a) (please see the attached file), b) (please see the attached file) where
Both (a) and (b) are false:
(please see the attached file)

Let:
(please see the attached file)

We see that (please see the attached file) but (please see the attached file) Now if (please see the attached file) then (please see the attached file) and (please see the attached file) But this implies that (please see the attached file) so (please see the attached file) can't be an integer. Thus (please see the attached file) is not closed under multiplication and so (please see the attached file) in not an ideal in (please see the attached file)

Let
(please see the attached file)

We see that (please see the attached file) but (please see the attached file) Now if (please see the attached file) then (please see the attached file) and (please see the attached file). But, this implies that (please see the attached file) so (please see the attached file) can't be an integer. Thus (please see the attached file) is not closed under multiplication and so (please see the attached file) in not an ideal in
(please see the attached file)

(please see the attached file) is ...

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