[proofplan] The trace and norm of $\beta \in L$ are the trace and determinant of the multiplication endomorphism $m_\beta: L \to L$, equivalently the negative coefficient of $x^{n-1}$ and $(-1)^n$ times the constant term of the characteristic polynomial $\chi_\beta(x) := \det(xI - m_\beta)$. By [Characteristic Polynomial via Minimal Polynomial](/theorems/1573), $\chi_\beta(x) = p_\beta(x)^{r}$ where $p_\beta$ is the minimal polynomial of $\beta$ over $\mathbb{Q}$ and $r := [L:\mathbb{Q}(\beta)]$. We identify the roots of $p_\beta$ in $\mathbb{C}$: these are precisely the distinct values $\beta_1, \ldots, \beta_m$ (where $m = \deg p_\beta$) taken by $\sigma(\beta)$ as $\sigma$ ranges over the $m$ embeddings of $\mathbb{Q}(\beta)$ into $\mathbb{C}$. Each embedding of $\mathbb{Q}(\beta)$ extends to exactly $r$ embeddings of $L$ (the [Embeddings of a Number Field](/theorems/1576) gives $r \cdot m = n$ total embeddings, partitioned into fibres of size $r$ over the embeddings of $\mathbb{Q}(\beta)$). Hence the multiset $\{\sigma_i(\beta) : i = 1, \ldots, n\}$ equals $\{\beta_1, \ldots, \beta_m\}$ with each $\beta_j$ repeated $r$ times. The roots of $\chi_\beta = p_\beta^r$ in $\mathbb{C}$ are therefore exactly this multiset. Computing the sum and product of these roots — the negative coefficient of $x^{n-1}$ and $(-1)^n$ the constant term of $\chi_\beta$ — produces the trace and norm formulas. [/proofplan] [step:Set notation for $\beta$, its minimal polynomial, and the characteristic polynomial of $m_\beta$] Let $n := [L:\mathbb{Q}]$. Let $\beta \in L$, set $m := [\mathbb{Q}(\beta):\mathbb{Q}]$ and $r := [L:\mathbb{Q}(\beta)]$, so the [tower law](/theorems/???) gives $n = r \cdot m$. Let $p_\beta \in \mathbb{Q}[x]$ be the monic minimal polynomial of $\beta$ over $\mathbb{Q}$; $\deg p_\beta = m$. Let $m_\beta: L \to L$ be the $\mathbb{Q}$-linear multiplication map $\gamma \mapsto \beta \gamma$, with characteristic polynomial $\chi_\beta(x) := \det(xI - m_\beta) \in \mathbb{Q}[x]$, monic of degree $n$. By definition of trace and norm, \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) &= \operatorname{tr}(m_\beta), \\ N_{L/\mathbb{Q}}(\beta) &= \det(m_\beta). \end{align*} Expressed in terms of $\chi_\beta(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_1 x + c_0$: \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = -c_{n-1}, \qquad N_{L/\mathbb{Q}}(\beta) = (-1)^n c_0. \tag{$\ast$} \end{align*} [guided] We fix the standing notation used throughout. - $L$ is a number field of degree $n$ over $\mathbb{Q}$, so $[L:\mathbb{Q}] = n < \infty$. - $\beta \in L$ is the element whose norm and trace we wish to compute. - $\mathbb{Q}(\beta) \subseteq L$ is the subfield generated by $\beta$. - $m := [\mathbb{Q}(\beta):\mathbb{Q}]$, $r := [L:\mathbb{Q}(\beta)]$. By the [tower law](/theorems/???), $n = rm$. - $p_\beta \in \mathbb{Q}[x]$ is the monic minimal polynomial of $\beta$ over $\mathbb{Q}$, of degree $m$. **The multiplication map and its characteristic polynomial**: the $\mathbb{Q}$-linear map \begin{align*} m_\beta: L &\to L, \\ \gamma &\mapsto \beta \gamma, \end{align*} is a $\mathbb{Q}$-linear endomorphism of the $n$-dimensional $\mathbb{Q}$-vector space $L$. Its characteristic polynomial \begin{align*} \chi_\beta(x) = \det(xI - m_\beta) \in \mathbb{Q}[x] \end{align*} is monic of degree $n$. By definition, \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = \operatorname{tr}(m_\beta), \qquad N_{L/\mathbb{Q}}(\beta) = \det(m_\beta). \end{align*} **Expressing trace and norm via coefficients of $\chi_\beta$**: for any $n \times n$ matrix $A$ with characteristic polynomial $\chi_A(x) = \det(xI - A) = x^n + c_{n-1} x^{n-1} + \cdots + c_0$, we have \begin{align*} \operatorname{tr}(A) = -c_{n-1}, \qquad \det(A) = (-1)^n c_0. \end{align*} (The first is the sum of eigenvalues, i.e. the sum of roots of $\chi_A$, i.e. $-c_{n-1}$ by Vieta. The second is the product of eigenvalues, which equals $(-1)^n$ times the constant term.) Applied here: \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = -c_{n-1}, \qquad N_{L/\mathbb{Q}}(\beta) = (-1)^n c_0. \end{align*} Our task is therefore to compute the sum and product of the roots of $\chi_\beta$ in $\mathbb{C}$. [/guided] [/step] [step:Express $\chi_\beta = p_\beta^r$ via [Characteristic Polynomial via Minimal Polynomial](/theorems/1573)] We apply [Characteristic Polynomial via Minimal Polynomial](/theorems/1573) with base field $K = \mathbb{Q}$, extension $L/\mathbb{Q}$, and element $\beta \in L$. Hypotheses: 1. **$L/\mathbb{Q}$ is a finite extension**: $[L:\mathbb{Q}] = n < \infty$ by the definition of a number field. 2. **$\beta \in L$ and $p_\beta$ is its minimal polynomial over $\mathbb{Q}$**: by construction. Both hold. The theorem concludes \begin{align*} \chi_\beta(x) = p_\beta(x)^r, \qquad r = [L:\mathbb{Q}(\beta)]. \tag{$\dagger$} \end{align*} [guided] We need to relate $\chi_\beta$ to something we can factor over $\mathbb{C}$ in a controlled way. The minimal polynomial $p_\beta$ is the natural candidate because it has a simple description via embeddings. The connection is given by [Characteristic Polynomial via Minimal Polynomial](/theorems/1573): for a finite extension $L/K$ and $\beta \in L$ with minimal polynomial $p_\beta \in K[x]$, the characteristic polynomial of $m_\beta: L \to L$ is $p_\beta^{[L:K(\beta)]}$. Verifying hypotheses with $K = \mathbb{Q}$: - **Finite extension $L/\mathbb{Q}$**: $[L:\mathbb{Q}] = n < \infty$. - **$p_\beta$ is the minimal polynomial of $\beta$ over $\mathbb{Q}$**: standing notation. Both hold; the theorem gives \begin{align*} \chi_\beta(x) = p_\beta(x)^r, \qquad \text{where } r = [L:\mathbb{Q}(\beta)]. \end{align*} Taking degrees: $\deg \chi_\beta = n$ should equal $r \cdot \deg p_\beta = rm$, consistent with $n = rm$ from the tower law. [/guided] [/step] [step:Identify the roots of $p_\beta$ in $\mathbb{C}$ with the values $\sigma(\beta)$ where $\sigma$ ranges over embeddings of $\mathbb{Q}(\beta)$] Let $\tau_1, \ldots, \tau_m: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}$ be the $m$ distinct field embeddings of $\mathbb{Q}(\beta)$ into $\mathbb{C}$ (their existence and count given by [Embeddings of a Number Field](/theorems/1576) applied to the number field $\mathbb{Q}(\beta)$ of degree $m$). Set \begin{align*} \beta_j := \tau_j(\beta), \qquad j = 1, \ldots, m. \end{align*} [claim:The $\beta_j$ are exactly the $m$ distinct roots of $p_\beta$ in $\mathbb{C}$] \begin{align*} \{\beta_1, \ldots, \beta_m\} &= \{z \in \mathbb{C} : p_\beta(z) = 0\}, \\ \beta_j \text{ are pairwise distinct}. \end{align*} [/claim] [proof] **Each $\beta_j$ is a root of $p_\beta$**: for each $j$, $\tau_j$ is a $\mathbb{Q}$-algebra homomorphism $\mathbb{Q}(\beta) \to \mathbb{C}$, so it commutes with polynomial expressions with $\mathbb{Q}$-coefficients. Applying $\tau_j$ to $p_\beta(\beta) = 0$: \begin{align*} p_\beta(\beta_j) = p_\beta(\tau_j(\beta)) = \tau_j(p_\beta(\beta)) = \tau_j(0) = 0. \end{align*} **The $\beta_j$ are pairwise distinct**: the proof of [Embeddings of a Number Field](/theorems/1576) established the bijection \begin{align*} \{\text{embeddings } \tau: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}\} \xrightarrow{\sim} \{\text{roots of } p_\beta \text{ in } \mathbb{C}\}, \qquad \tau \mapsto \tau(\beta). \end{align*} Distinct embeddings correspond to distinct roots, so $\beta_i \neq \beta_j$ for $i \neq j$. **All roots arise this way**: by [Embeddings of a Number Field](/theorems/1576), $p_\beta$ (monic of degree $m$, irreducible over $\mathbb{Q}$, hence separable because $\mathbb{Q}$ has characteristic $0$) has exactly $m$ distinct roots in $\mathbb{C}$; and we have produced $m$ distinct roots $\beta_1, \ldots, \beta_m$ already. Hence $\{\beta_j\}_{j=1}^m$ exhausts the set of complex roots of $p_\beta$. [/proof] [guided] We now describe the roots of $p_\beta$ in $\mathbb{C}$ using embeddings — this is where [Embeddings of a Number Field](/theorems/1576) enters. **Apply the embeddings theorem to the number field $\mathbb{Q}(\beta)$.** The subfield $\mathbb{Q}(\beta) \subseteq L$ is itself a number field (a finite extension of $\mathbb{Q}$, of degree $m$). By [Embeddings of a Number Field](/theorems/1576), it has exactly $m$ field embeddings into $\mathbb{C}$. List them: \begin{align*} \tau_1, \ldots, \tau_m: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}. \end{align*} Set \begin{align*} \beta_j := \tau_j(\beta), \qquad j = 1, \ldots, m. \end{align*} **The $\beta_j$ are roots of $p_\beta$.** Each $\tau_j$ is a $\mathbb{Q}$-algebra homomorphism, so it commutes with polynomial expressions with $\mathbb{Q}$-coefficients: \begin{align*} p_\beta(\beta_j) = p_\beta(\tau_j(\beta)) = \tau_j(p_\beta(\beta)) = \tau_j(0) = 0. \end{align*} So each $\beta_j$ is a root of $p_\beta$ in $\mathbb{C}$. **The $\beta_j$ are pairwise distinct.** The proof of [Embeddings of a Number Field](/theorems/1576) constructed an explicit bijection: \begin{align*} \{\text{embeddings } \tau: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}\} \xrightarrow{\sim} \{\text{roots of } p_\beta \text{ in } \mathbb{C}\}, \qquad \tau \mapsto \tau(\beta). \end{align*} Distinct embeddings $\tau_i, \tau_j$ correspond to distinct roots $\beta_i, \beta_j$ under this bijection. So the list $\beta_1, \ldots, \beta_m$ has $m$ distinct elements. **These are all the roots.** The polynomial $p_\beta$ is monic of degree $m$, irreducible in $\mathbb{Q}[x]$ (as the minimal polynomial), hence separable over $\mathbb{Q}$ because $\operatorname{char}\mathbb{Q} = 0$. By [Embeddings of a Number Field](/theorems/1576) (or more precisely its underlying fact: separable irreducible polynomials of degree $m$ have exactly $m$ distinct roots in any algebraically closed extension), $p_\beta$ has exactly $m$ distinct roots in $\mathbb{C}$. Since we have exhibited $m$ distinct roots, $\{\beta_1, \ldots, \beta_m\}$ is the full set. **Conclusion of this step**: \begin{align*} p_\beta(x) = \prod_{j=1}^m (x - \beta_j) \qquad \text{in } \mathbb{C}[x], \end{align*} with $\beta_j = \tau_j(\beta)$ and the $\beta_j$ all distinct. These are the "conjugates" of $\beta$ in $\mathbb{C}$ up to relabelling. [/guided] [/step] [step:Partition the $n$ embeddings $L \hookrightarrow \mathbb{C}$ into fibres of size $r$ over the $m$ embeddings of $\mathbb{Q}(\beta)$] Let $\sigma_1, \ldots, \sigma_n: L \hookrightarrow \mathbb{C}$ be the $n$ distinct field embeddings of $L$ into $\mathbb{C}$ (given by [Embeddings of a Number Field](/theorems/1576) applied to $L$, whose hypotheses — namely that $L$ is a number field — hold by standing assumption). Each $\sigma_i$ restricts to a field embedding $\sigma_i|_{\mathbb{Q}(\beta)}: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}$, which must equal one of $\tau_1, \ldots, \tau_m$. This defines a restriction map \begin{align*} \rho: \{\sigma_1, \ldots, \sigma_n\} &\to \{\tau_1, \ldots, \tau_m\}, \\ \sigma_i &\mapsto \sigma_i|_{\mathbb{Q}(\beta)}. \end{align*} [claim:Each fibre of $\rho$ has exactly $r$ elements] For every $j \in \{1, \ldots, m\}$, $|\rho^{-1}(\tau_j)| = r$. [/claim] [proof] Fix $\tau_j$. Composing $\tau_j$ with the inclusion $\mathbb{C} \hookrightarrow \mathbb{C}$ gives a $\mathbb{Q}$-algebra homomorphism $\tau_j: \mathbb{Q}(\beta) \to \mathbb{C}$. The elements of $\rho^{-1}(\tau_j)$ are exactly the $\mathbb{Q}$-algebra homomorphisms $\sigma: L \to \mathbb{C}$ with $\sigma|_{\mathbb{Q}(\beta)} = \tau_j$, equivalently, the $\mathbb{Q}(\beta)$-algebra homomorphisms $\sigma: L \to \mathbb{C}$ when $\mathbb{C}$ is given the $\mathbb{Q}(\beta)$-algebra structure via $\tau_j$. We count these. $L$ is a finite separable extension of $\mathbb{Q}(\beta)$ (finite because $[L:\mathbb{Q}(\beta)] = r < \infty$; separable because any extension of a characteristic-$0$ field is separable). Applying the general fact — the same counting argument underlying [Embeddings of a Number Field](/theorems/1576), but relative to the base field $\mathbb{Q}(\beta)$ instead of $\mathbb{Q}$ — the number of $\mathbb{Q}(\beta)$-algebra homomorphisms $L \to \mathbb{C}$ (where $\mathbb{C}$ is a $\mathbb{Q}(\beta)$-algebra via any fixed embedding, here $\tau_j$) equals $[L:\mathbb{Q}(\beta)] = r$. Hence $|\rho^{-1}(\tau_j)| = r$ for each $j$. [/proof] In particular, \begin{align*} n = |\{\sigma_1, \ldots, \sigma_n\}| = \sum_{j=1}^m |\rho^{-1}(\tau_j)| = \sum_{j=1}^m r = rm, \end{align*} reconfirming $n = rm$. [guided] We now count, in the opposite direction, the $n$ embeddings of $L$ by their restrictions to $\mathbb{Q}(\beta)$. **The $n$ embeddings of $L$**: by [Embeddings of a Number Field](/theorems/1576), the number field $L/\mathbb{Q}$ of degree $n$ has exactly $n$ field embeddings $L \hookrightarrow \mathbb{C}$. Enumerate them $\sigma_1, \ldots, \sigma_n$. **Restriction to $\mathbb{Q}(\beta)$**: each $\sigma_i: L \hookrightarrow \mathbb{C}$, restricted to the subfield $\mathbb{Q}(\beta) \subseteq L$, gives a field embedding $\sigma_i|_{\mathbb{Q}(\beta)}: \mathbb{Q}(\beta) \hookrightarrow \mathbb{C}$. Since there are exactly $m$ embeddings of $\mathbb{Q}(\beta)$ into $\mathbb{C}$ (namely $\tau_1, \ldots, \tau_m$), the restriction $\sigma_i|_{\mathbb{Q}(\beta)}$ equals some $\tau_j$. This defines a **restriction map** \begin{align*} \rho: \{\sigma_1, \ldots, \sigma_n\} \to \{\tau_1, \ldots, \tau_m\}, \qquad \sigma \mapsto \sigma|_{\mathbb{Q}(\beta)}. \end{align*} **Counting the fibres.** We claim each $\tau_j$ has exactly $r$ preimages under $\rho$. Fix $\tau_j$. The preimages $\sigma \in \rho^{-1}(\tau_j)$ are precisely the $\mathbb{Q}$-algebra embeddings $\sigma: L \hookrightarrow \mathbb{C}$ whose restriction to $\mathbb{Q}(\beta)$ is $\tau_j$. Equivalently — if we give $\mathbb{C}$ a $\mathbb{Q}(\beta)$-algebra structure via $\tau_j: \mathbb{Q}(\beta) \to \mathbb{C}$ (this makes $\mathbb{C}$ into a $\mathbb{Q}(\beta)$-algebra with structure map $\tau_j$) — they are precisely the $\mathbb{Q}(\beta)$-algebra homomorphisms $L \to \mathbb{C}$. Now count these. $L$ is a finite extension of $\mathbb{Q}(\beta)$, of degree $r = [L:\mathbb{Q}(\beta)]$, and it is separable over $\mathbb{Q}(\beta)$ (in characteristic $0$, every algebraic extension is separable). By the same argument that underlies [Embeddings of a Number Field](/theorems/1576) but now with base field $\mathbb{Q}(\beta)$ instead of $\mathbb{Q}$ (the proof of that theorem works verbatim over any field of characteristic $0$): the number of $\mathbb{Q}(\beta)$-algebra homomorphisms $L \to \mathbb{C}$ equals $[L:\mathbb{Q}(\beta)] = r$. Hence $|\rho^{-1}(\tau_j)| = r$ for each $j \in \{1, \ldots, m\}$. **Sanity check**: summing, \begin{align*} n = |\{\sigma_i\}| = \sum_{j=1}^m |\rho^{-1}(\tau_j)| = \sum_{j=1}^m r = rm, \end{align*} consistent with the tower law. **Geometric picture**: the $n$ embeddings of $L$ are organised into $m$ groups of size $r$; all $\sigma$'s in group $j$ send $\beta$ to the same complex number $\tau_j(\beta) = \beta_j$, but they differ in how they send elements of $L$ not in $\mathbb{Q}(\beta)$. [/guided] [/step] [step:Compute the multiset $\{\sigma_i(\beta)\}_{i=1}^n$ and factor $\chi_\beta$ over $\mathbb{C}$] For any embedding $\sigma: L \hookrightarrow \mathbb{C}$ with $\rho(\sigma) = \tau_j$, the restriction $\sigma|_{\mathbb{Q}(\beta)} = \tau_j$, so in particular $\sigma(\beta) = \tau_j(\beta) = \beta_j$. Therefore the multiset of values is \begin{align*} \{\!\{\sigma_i(\beta) : i = 1, \ldots, n\}\!\} = \{\!\{\beta_j \text{ with multiplicity } r : j = 1, \ldots, m\}\!\}. \tag{$\ddagger$} \end{align*} On the other hand, from the factorisations $p_\beta(x) = \prod_{j=1}^m (x - \beta_j)$ in $\mathbb{C}[x]$ (from the claim in the previous step) and $\chi_\beta = p_\beta^r$ (from $(\dagger)$): \begin{align*} \chi_\beta(x) = p_\beta(x)^r = \prod_{j=1}^m (x - \beta_j)^r. \end{align*} So the multiset of roots of $\chi_\beta$ in $\mathbb{C}$, counted with multiplicity, is exactly the multiset on the right-hand side of $(\ddagger)$. In particular, combining these two descriptions of the same multiset, \begin{align*} \chi_\beta(x) = \prod_{i=1}^n (x - \sigma_i(\beta)). \tag{$\star$} \end{align*} [guided] We now connect the fibre partition of $\{\sigma_i\}$ to the factorisation of $\chi_\beta$ over $\mathbb{C}$. **Values of $\sigma_i$ at $\beta$.** For any $\sigma \in \rho^{-1}(\tau_j)$, by definition $\sigma|_{\mathbb{Q}(\beta)} = \tau_j$, so in particular $\sigma(\beta) = \tau_j(\beta) = \beta_j$. In other words: every embedding in the fibre over $\tau_j$ sends $\beta$ to the common value $\beta_j$. Since $|\rho^{-1}(\tau_j)| = r$ for each $j$, the multiset of values \begin{align*} \{\!\{\sigma_1(\beta), \ldots, \sigma_n(\beta)\}\!\} \end{align*} consists of $\beta_1$ (with multiplicity $r$), $\beta_2$ (with multiplicity $r$), ..., $\beta_m$ (with multiplicity $r$). We write this as \begin{align*} \{\!\{\sigma_i(\beta) : i = 1, \ldots, n\}\!\} = \{\!\{\beta_1, \ldots, \beta_1, \beta_2, \ldots, \beta_2, \ldots, \beta_m, \ldots, \beta_m\}\!\}, \end{align*} each $\beta_j$ repeated $r$ times, for a total of $rm = n$ entries. **Factorisation of $\chi_\beta$ over $\mathbb{C}$.** From the previous steps: - $p_\beta(x) = \prod_{j=1}^m (x - \beta_j)$ in $\mathbb{C}[x]$ (factorisation over the algebraic closure, by distinctness of the $\beta_j$). - $\chi_\beta(x) = p_\beta(x)^r$ in $\mathbb{Q}[x] \subseteq \mathbb{C}[x]$ (from the relation $(\dagger)$). Combining: \begin{align*} \chi_\beta(x) = \left[ \prod_{j=1}^m (x - \beta_j) \right]^r = \prod_{j=1}^m (x - \beta_j)^r. \end{align*} **Two descriptions of the same multiset of roots.** The roots of $\chi_\beta$ in $\mathbb{C}$, counted with multiplicity, form the multiset $\{\!\{\beta_j \text{ with multiplicity } r\}\!\}_{j=1}^m$. But this is the same multiset as $\{\!\{\sigma_i(\beta)\}\!\}_{i=1}^n$. Therefore we can reassemble: \begin{align*} \chi_\beta(x) = \prod_{i=1}^n (x - \sigma_i(\beta)). \end{align*} This is the key identity: the characteristic polynomial of $m_\beta$ factors over $\mathbb{C}$ as the product of linear factors indexed by the $n$ embeddings of $L$, each contributing a root $\sigma_i(\beta)$. [/guided] [/step] [step:Extract trace and norm from the factorisation] Expand $(\star)$: \begin{align*} \chi_\beta(x) = \prod_{i=1}^n (x - \sigma_i(\beta)) = x^n - \left( \sum_{i=1}^n \sigma_i(\beta) \right) x^{n-1} + \cdots + (-1)^n \prod_{i=1}^n \sigma_i(\beta). \end{align*} Comparing with $\chi_\beta(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_0$: \begin{align*} c_{n-1} = - \sum_{i=1}^n \sigma_i(\beta), \qquad c_0 = (-1)^n \prod_{i=1}^n \sigma_i(\beta). \end{align*} Substituting into $(\ast)$: \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = -c_{n-1} = \sum_{i=1}^n \sigma_i(\beta), \qquad N_{L/\mathbb{Q}}(\beta) = (-1)^n c_0 = (-1)^n \cdot (-1)^n \prod_{i=1}^n \sigma_i(\beta) = \prod_{i=1}^n \sigma_i(\beta), \end{align*} using $(-1)^{2n} = 1$. This proves the stated formulas, completing the proof. [guided] The final step is pure bookkeeping. Expand the factorisation $(\star)$: \begin{align*} \chi_\beta(x) = \prod_{i=1}^n (x - \sigma_i(\beta)). \end{align*} Using Vieta's formulas (or, equivalently, direct expansion and matching coefficients): \begin{align*} \chi_\beta(x) &= x^n - \left( \sum_{i=1}^n \sigma_i(\beta) \right) x^{n-1} + \left(\sum_{i < j} \sigma_i(\beta)\sigma_j(\beta)\right) x^{n-2} - \cdots + (-1)^n \prod_{i=1}^n \sigma_i(\beta). \end{align*} Writing $\chi_\beta(x) = x^n + c_{n-1} x^{n-1} + \cdots + c_0$, we read off: \begin{align*} c_{n-1} &= -\sum_{i=1}^n \sigma_i(\beta), \\ c_0 &= (-1)^n \prod_{i=1}^n \sigma_i(\beta). \end{align*} Substituting into $(\ast)$ from the first step: \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = -c_{n-1} = -\left(-\sum_{i=1}^n \sigma_i(\beta)\right) = \sum_{i=1}^n \sigma_i(\beta). \end{align*} And: \begin{align*} N_{L/\mathbb{Q}}(\beta) = (-1)^n c_0 = (-1)^n \cdot (-1)^n \prod_{i=1}^n \sigma_i(\beta) = \prod_{i=1}^n \sigma_i(\beta), \end{align*} where we used $(-1)^n \cdot (-1)^n = (-1)^{2n} = 1$. This gives the desired formulas: \begin{align*} \operatorname{tr}_{L/\mathbb{Q}}(\beta) = \sum_{i=1}^n \sigma_i(\beta), \qquad N_{L/\mathbb{Q}}(\beta) = \prod_{i=1}^n \sigma_i(\beta), \end{align*} and the values $\sigma_1(\beta), \ldots, \sigma_n(\beta)$ are the **conjugates** of $\beta$ in $\mathbb{C}$, partitioned as $\{\beta_1, \ldots, \beta_1, \beta_2, \ldots, \beta_2, \ldots, \beta_m, \ldots, \beta_m\}$ with each $\beta_j$ repeated $r$ times. **A structural remark**: when $\beta$ is a primitive element, i.e. $L = \mathbb{Q}(\beta)$, then $r = 1$ and $m = n$, and the conjugates are simply the $n$ distinct roots of $p_\beta$ in $\mathbb{C}$. When $\beta$ is a "sub-primitive" element generating a proper subfield $\mathbb{Q}(\beta) \subsetneq L$, then $r > 1$ and the conjugates of $\beta$ come with repetitions: each distinct value $\beta_j$ appears exactly $r$ times in the list $(\sigma_1(\beta), \ldots, \sigma_n(\beta))$. The trace and norm formulas automatically take this multiplicity into account because they sum/multiply over all $n$ embeddings, not over the distinct roots of $p_\beta$. [/guided] [/step]