Isomorphism of rings question? 10 pts for informative answer!

Hello Experts,

How do I prove that the kernel of phi: Z[sqrt(3)] -------> Z_11

is the ideal generated by: <11,-5+sqrt(3)>

Please give a step by step answer.

How are you defining phi?

Once this is clear, then I should be able to answer your question.

You make it hard to edit if you prematurely give "best answer" (I had the solution all written up just at this point). Here it is again...

φ is onto.
Given x in Z_11:
As x = n (mod 11) for some n = 0, 1, 2, ..., 10,
n is in Z[√3], and φ(n) = n = x (mod 11) as required.

kernel is in next comment.

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ker φ = {a+b√3: φ(a+b√3) = 0 mod 11}
= {a+b√3: a+5b = 0 mod 11}
= {a+b√3: a = -5b + 11m for some m in Z}
= {-5b + 11k + b√3: b, m in Z}
= {b(-5+ √3) + 11m: b, m in Z}
= <-5+ √3, 11>.

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