Silly mistake in this number theory book

Question

My question is very easily to be solved (at least I hope so)

I think this book has a mistake:

When I calculate I get $b_3\equiv -2 (\mod{2})$ which implies $b_3=0$, am I right?

Another question, why $I_1I_2=(3)$? following my calculations we have $I_1I_2=(3)\big(1,\frac{1+2\sqrt{10}}{2}\big)$ if $b_3=1$ and $I_1I_2=(3)\big(1,\sqrt{10}\big)$ if $b_3=0$.

Thanks in advance

Answer 1

You are right. I also get the same congruence for $b_3$ as you did. Your second question is easy. Because the ideal $(1,\sqrt{10})$ contains a unit, it is equal to the whole ring (of integers) $R$, i.e. $I_1I_2=(3)R=(3)$. Here $R=\mathbb{Z}[\sqrt{10}]$.

Answer 2

Simpler: $\ (3,w)(3,w') = (3(3,w'),w(3,w')) = (9,3w',3w,9) = 3(3,w,w') = (3)\ $ since $\,(3,w,w')\supseteq (3,w\!-\!w') = (3,2) = (1).$