Calculating norms of prime ideal by hand

Question

Let $K=\mathbb{Q}(a)$ with $a^3+10a+1=0$. Note that $\mathfrak{p}=(2,1+a)$ is a prime ideal. In my course notes, it says that $N\mathfrak{p}=2$, where $N I:= (\mathcal{O}_K:I)$ for any ideal $I$ of $\mathcal{O}_K=\mathbb{Z}[a]$, the ring of integers over $K$. My question is, how to go about calculating $N \mathfrak{p}$ without using a computer? Is there a systematic way? In other words, I don't know how to obtain the answer of 2 by hand. Thanks for any help.

Answer 1

Note that $\Bbb Z[a] \cong \Bbb Z[X]/(X^3 + 10X + 1)$. Thus, the following holds;

\begin{align} \Bbb Z[a]/(2, 1+a) &\cong \Bbb Z[X]/(2, 1 + X, X^3 + 10X + 1)\\ &\cong \Bbb F_2[X]/(1+ X, X^3 + 1)\\ &\cong \Bbb F_2[X]/(1 + X)\\ &\cong \Bbb F_2. \end{align}

Now we have that $(\mathcal{O}_K:\mathfrak{p}) = 2.$