1. ## Ideal Question....

Dealing with Z[Sqrt(-13)], let J = (7, -1+Sqrt(-13) ). ie - J is the ideal generate by 7 & -1+Sqrt(-13)

Find an integer N s.t. J^2 = (49, n+Sqrt(-13) )

Many thanks in advance. :-) x

2. Originally Posted by AAM
Dealing with Z[Sqrt(-13)], let J = (7, -1+Sqrt(-13) ). ie - J is the ideal generate by 7 & -1+Sqrt(-13)

Find an integer N s.t. J^2 = (49, n+Sqrt(-13) )
Not being an algebraist, I'm not 100% sure of this, but it looks to me as though J^2 should contain the elements $7^2=49$, $(-1+\sqrt{-13})^2 = -12-2\sqrt{-13}$ and $7(-1+\sqrt{-13}) = -7+7\sqrt{-13}$. Therefore it will also contain $49 + 3(-12-2\sqrt{-13}) + (-7+7\sqrt{-13}) = 6 + \sqrt{-13}$. So I would go for n=6.

3. Awesome - thank you! :-)

Is there a more efficient way of finding n, or do you just have to "play around with numbers" as you have done? :-)