# Norm and Trace ## Two Fundamental Invariants Let $K$ be a number field of degree $$ [K:\mathbb{Q}]=n. $$ Each element $\alpha\in K$ has two fundamental rational invariants: its norm and its trace. These generalize familiar operations from quadratic fields. The norm behaves like a product of conjugates. The trace behaves like a sum of conjugates. If the embeddings of $K$ into $\mathbb{C}$ are $$ \sigma_1,\sigma_2,\ldots,\sigma_n, $$ then the norm and trace are defined by $$ N_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)\sigma_2(\alpha)\cdots\sigma_n(\alpha), $$ and $$ \operatorname{Tr}_{K/\mathbb{Q}}(\alpha) = \sigma_1(\alpha)+\sigma_2(\alpha)+\cdots+\sigma_n(\alpha). $$ These quantities belong to $\mathbb{Q}$. If $\alpha$ is an algebraic integer, then both belong to $\mathbb{Z}$. ## Quadratic Fields Let $$ K=\mathbb{Q}(\sqrt d), $$ where $d$ is squarefree. Every element has the form $$ \alpha=a+b\sqrt d, \qquad a,b\in\mathbb{Q}. $$ There are two embeddings into $\mathbb{C}$: $$ \sqrt d\mapsto \sqrt d, \qquad \sqrt d\mapsto -\sqrt d. $$ Hence the conjugate of $\alpha$ is $$ \overline{\alpha}=a-b\sqrt d. $$ The norm is $$ N(\alpha) = \alpha\overline{\alpha} = (a+b\sqrt d)(a-b\sqrt d) = a^2-db^2. $$ The trace is $$ \operatorname{Tr}(\alpha) = \alpha+\overline{\alpha} = 2a. $$ These formulas are the simplest model for the general theory. ## Example: Gaussian Integers In the Gaussian field $$ \mathbb{Q}(i), $$ an element has the form $$ \alpha=a+bi. $$ Its conjugate is $$ \overline{\alpha}=a-bi. $$ Therefore $$ N(a+bi)=a^2+b^2, $$ and $$ \operatorname{Tr}(a+bi)=2a. $$ The norm explains why sums of two squares appear naturally in Gaussian integer arithmetic. For example, $$ N(3+4i)=3^2+4^2=25. $$ Since the norm is multiplicative, $$ N(\alpha\beta)=N(\alpha)N(\beta), $$ the product of two sums of squares is again a sum of squares. ## Norm as Determinant There is another useful definition of the norm. Multiplication by $\alpha$ defines a $\mathbb{Q}$-linear map $$ m_\alpha:K\to K, \qquad x\mapsto \alpha x. $$ Since $K$ is an $n$-dimensional vector space over $\mathbb{Q}$, this map has a determinant. One has $$ N_{K/\mathbb{Q}}(\alpha)=\det(m_\alpha). $$ Similarly, $$ \operatorname{Tr}_{K/\mathbb{Q}}(\alpha)=\operatorname{trace}(m_\alpha). $$ This linear algebra viewpoint is often the cleanest definition in abstract settings. ## Multiplicativity and Additivity The norm is multiplicative: $$ N(\alpha\beta)=N(\alpha)N(\beta). $$ This follows either from the embedding definition or from the determinant interpretation. The trace is additive: $$ \operatorname{Tr}(\alpha+\beta) = \operatorname{Tr}(\alpha)+\operatorname{Tr}(\beta). $$ It is also compatible with rational scaling: $$ \operatorname{Tr}(q\alpha)=q\operatorname{Tr}(\alpha) $$ for $q\in\mathbb{Q}$. Thus the norm behaves like a multiplicative size, while the trace behaves like a linear measurement. ## Minimal Polynomials Suppose $\alpha$ has minimal polynomial $$ m_\alpha(x) = x^r+c_{r-1}x^{r-1}+\cdots+c_1x+c_0. $$ If $K=\mathbb{Q}(\alpha)$, then the conjugates of $\alpha$ are the roots of this polynomial. Thus $$ \operatorname{Tr}_{K/\mathbb{Q}}(\alpha)=-c_{r-1}, $$ and $$ N_{K/\mathbb{Q}}(\alpha)=(-1)^r c_0. $$ For example, $\sqrt2$ has minimal polynomial $$ x^2-2. $$ The trace is $$ 0, $$ and the norm is $$ -2. $$ Indeed, $$ N(\sqrt2)=(\sqrt2)(-\sqrt2)=-2. $$ ## Algebraic Integers If $$ \alpha\in\mathcal O_K, $$ then all conjugates of $\alpha$ are algebraic integers. Their sum and product are rational algebraic integers, hence ordinary integers. Therefore $$ N_{K/\mathbb{Q}}(\alpha)\in\mathbb{Z}, \qquad \operatorname{Tr}_{K/\mathbb{Q}}(\alpha)\in\mathbb{Z}. $$ This makes norm and trace useful arithmetic tools. For example, if $\alpha\in\mathcal O_K$ and $$ N(\alpha)=\pm1, $$ then $\alpha$ is a unit. ## Norms and Divisibility Norms often reduce divisibility questions in $\mathcal O_K$ to divisibility questions in $\mathbb{Z}$. If $$ \alpha\mid\beta $$ in $\mathcal O_K$, then $$ N(\alpha)\mid N(\beta) $$ in $\mathbb{Z}$, up to sign. This gives a useful obstruction. For example, in the Gaussian integers, if $$ \alpha\mid 5, $$ then $$ N(\alpha)\mid N(5)=25. $$ Thus possible norms of divisors are restricted to divisors of $25$. ## Trace Pairing The trace defines a bilinear form on $K$: $$ (\alpha,\beta)\mapsto \operatorname{Tr}_{K/\mathbb{Q}}(\alpha\beta). $$ This form is fundamental in the study of discriminants and integral bases. Given a basis $$ \omega_1,\ldots,\omega_n $$ of $K$ over $\mathbb{Q}$, one forms the matrix $$ \left(\operatorname{Tr}(\omega_i\omega_j)\right)_{i,j}. $$ Its determinant is the discriminant of the basis. Thus trace connects field arithmetic with lattice geometry. ## Geometric Meaning Under the embeddings $$ \sigma_i:K\to\mathbb{C}, $$ an element $\alpha$ becomes a tuple of complex numbers: $$ (\sigma_1(\alpha),\ldots,\sigma_n(\alpha)). $$ The norm is their product. The trace is their sum. When restricted to the ring of integers, these embeddings place $\mathcal O_K$ as a lattice in Euclidean space. Norm and trace then measure arithmetic information through this geometric embedding. ## Arithmetic Role Norm and trace are among the most important tools in algebraic number theory. They are used to study: - units, - divisibility, - ideals, - discriminants, - integral bases, - ramification, - field extensions. The norm converts multiplication in a number field into multiplication in $\mathbb{Q}$. The trace converts addition and linear structure in a number field into ordinary rational data. Together, they provide the basic bridge between arithmetic inside $K$ and arithmetic over $\mathbb{Q}$.