At the moment I am learning algebraic number theory and I have seemed to be missing some basic understanding.
How do you multiply ideals
For example
$$(2,1+\sqrt{-5})^2 = (2,1+\sqrt{-5})(2,1+\sqrt{-5}) = (4, 2+2\sqrt{-5}, -4+2\sqrt{-5}) $$
but the book says
$$ (2, 2+2\sqrt{-5}, -4+2\sqrt{-5}) $$
why? my understanding was that you multiply each of the generators in one bracket with the other bracket and ignore duplicates or multiples.
can someone help please, please also give some worked examples that I may work on to understand.
thank you in advance.
H
$(2+2\sqrt{-5})-(-4+2\sqrt{-5})=6$, so $2$ is in the ideal iff $4$ is in the ideal. They are the same set.