I need to prove $(23,\alpha-10)$ is a prime ideal in $Z[\alpha]/(\alpha^3-\alpha-1)$. I want to show that $Norm(23,\alpha-10)$ is a prime number, thus proving it is a prime as we are in a Dedekind domain. Can anyone help me to compute the ideal norm?
Also, in particular, I know that for a PID, $R$, $Norm(I)$, $I\in R$ is given by the generator conjugate's products. Can anyone help me to get the connection between this definition and $Norm(I)=|R/I|$?
On
Let's label $R=\mathbb{Z}[\alpha]/\langle\alpha^3-\alpha-1\rangle$ and $I=\langle 23,\alpha-10 \rangle$. Then we can identify (as abelian groups) $$R\cong \mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$$ via the map $x+y\alpha+z\alpha^2\mapsto (x,y,z)$. Note that this map identifies (again as abelian groups) $$I\cong \langle(23, 0, 0), (-10, 1, 0), (-100, 0, 1)\rangle$$ (you should check this to be sure for yourself!). At this point the quotient (as an abelian group) on the right hand side of this identification is clear, it isomorphic to $\mathbb{Z}/23\mathbb{Z}$.
Since ${\rm disc}(x^3-x-1) = -23$ is squarefree, the ring of integers of the field $\mathbf Q(\alpha)$ is $\mathbf Z[\alpha]$ (do you know why?), so the way an ideal $(p) = p\mathbf Z[\alpha]$ decomposes in $\mathbf Z[\alpha]$ matches how $x^3 - x - 1$ decomposes in $(\mathbf Z/(p))[x]$. Is this a result you've heard of already? Since $x^3 - x - 1 \equiv (x -3)(x - 10)^2 \bmod 23$, we have the prime ideal factorization $(23) = (23, \alpha-3)(23,\alpha-10)^2$ where both prime ideals $(23, \alpha-3)$ and $(23,\alpha-10)$ have norm $23^1$ since both irreducible factors of $x^3 - x - 1 \bmod 23$ have degree $1$.
More generally, the Dedekind--Kummer theorem says that in every number field $K$ with ring of integers $\mathcal O_K$, if $\alpha$ is an algebraic integer generating $K$ over $\mathbf Q$ then (even if $\mathcal O_K \not= \mathbf Z[\alpha]$) for each prime $p$ not dividing $[\mathcal O_K:\mathbf Z[\alpha]]$, the way $(p) = p\mathcal O_K$ decomposes into prime ideals in $\mathcal O_K$ resembles how $f(x)$ decomposes in $(\mathbf Z/(p))[x]$ where $f(x)$ is the minimal polynomial of $\alpha$ over $\mathbf Q$ (necessarily $f(x)$ is monic in $\mathbf Z[x]$): if $$ f(x) \equiv \pi_1(x)^{e_1} \cdots \pi_g(x)^{e_g} \bmod p $$ with irreducible $\pi_i(x)$ in $(\mathbf Z/(p))[x]$ then $$ p\mathbf Z[\alpha] = \mathfrak p_1^{e_1} \cdots \mathfrak p_g^{e_g} $$ where $\mathfrak p_i = (p,\pi_i(\alpha))$ and the norm of $\mathfrak p_i$ is $p^{\deg \pi_i}$. (Although $\pi_i(x)$ is defined as a polynomial in $(\mathbf Z/(p))[x]$, the ideal $(p,\pi_i(\alpha))$ is well-defined because the ambiguity of the lift of $\pi_i(x)$ to $\mathbf Z[x]$ is washed out in the ideal $(p,\pi_i(\alpha))$ since the ideal contains $p$.) See here for further details.
Your description of the norm of a principal ideal as a product of conjugates of a generators is not the right way to think about the norm as a foundational concept. You should define the field norm ${\rm N}_{K/\mathbf Q}(\alpha)$ of an element $\alpha$ in a number field $K$ as $\det(m_\alpha)$ where $m_\alpha \colon K \to K$ is the multiplication-by-$\alpha$ map viewed as a $\mathbf Q$-linear map. That this definition for ${\rm N}_{K/\mathbf Q}(\alpha)$ agrees with the "product of conjugates" viewpoint is here (see Theorem 5.9 and Corollary 5.15 to relate minimal and characteristic polynomials of an element in a finite extension field). For every nonzero ideal $\mathfrak a$ in $\mathcal O_K$, the ideal norm ${\rm N}(\mathfrak a)$ of $\mathfrak a$ is defined to to be the index $[\mathcal O_K:\mathfrak a]$. The compatibility between these two concepts of norm (of an element and an ideal) occurs in the case where $\mathfrak a = (\alpha)$ is a principal ideal for some nonzero $\alpha \in \cal O_K$, where the compatibility is ${\rm N}((\alpha)) = |{\rm N}_{K/\mathbf Q}(\alpha)|$. That formula is a special case of Theorem 5.15 here by taking $M = \mathcal O_K$, $M' = \alpha\mathcal O_K$, and letting $(c_{ij})$ be the matrix expressing a $\mathbf Z$-basis of $\alpha\mathcal O_K$ in terms of a $\mathbf Z$-basis of $\mathcal O_K$. Moreover, that also gives an approach to computing the norm of an arbitrary nonzero ideal $\mathfrak a$ in $\mathcal O_K$: ${\rm N}(\mathfrak a) = |\det(c_{ij})|$ where $(c_{ij})$ is the matrix describing a $\mathbf Z$-basis of $\mathfrak a$ in terms of a $\mathbf Z$-basis of $\mathcal O_K$. A completely different method of computing ${\rm N}(\mathfrak a)$ by using a field norm for $K(T)/\mathbf Q(T)$, due to Kronecker, is described in the top MO answer here.