What is the relation between order in a number field and order in a quaternion algebra?

Question

The study of number field theory and quaternion algebra theory confuses me a lot: there are always defined symmetrical quantities (for example the trace, the norm, the orders). But I can't understand the underlying link.

• Is there a correspondence of some kind between the norm of an element in a number field and the norm in a quaternion algebra?
• Is there a correspondence of some kind between the trace of an element in a number field and the trace in a quaternion algebra?
• Is there a correspondence of some kind between an order in a number field and an order in a quaternion algebra?

Thanks for the help.

Answer 1

The (reduced) norm, (reduced) trace, (reduced) characteristic polynomial and orders have generalizations to general finite-dimensional (unitary, associative) algebras over a field.

The principal invariants. Take a finite-dimensional algebra $A$ over a field $K$, and choose a basis $(e_i)$. The principal polynomial of $A$ (w.r.t. the basis) is the minimal polynomial of the "generic" element $\sum x_i e_i \in A \otimes_K K(x_1, \ldots, x_n)$. You can specialize it by plugging in values $\lambda_i \in K$ for every $x_i$, and you then call this the principal polynomial of the element $a = \sum \lambda_i e_i$. Denote it by $P_{A, a}$. (It does not depend on the choice of basis.) When $A$ is a field extension of $K$, the principal polynomial is just the usual characteristic polynomial. When $A$ is a quaternion algebra (or more generally, a central simple algebra) the principal polynomial is the reduced characteristic polynomial. One can show that $$P_{A, a}(T) = \prod_{ (\rho, V) \in \operatorname{Irr}(A \otimes_K \overline K)} \chi_{\rho(a)}(T) \,,$$ where the product runs over the isomorphism classes of simple left $A \otimes_{K} \overline K$-modules, and $\chi_b(T)$ denotes the characteristic polynomial of $b \in \operatorname{End}(V)$. (See Jacobson's The Theory of Rings, Chapter 5 §18.)

Now define the principal trace of $a \in A$ to be minus the coefficient of $T^{\deg P_A - 1}$ in $P_{A, a}$, and define the principal norm of $a \in A$ to be $(-1)^{\deg P_A}$ times the constant term of $P_{A, a}$. The above expression for $P_{A, a}$ implies that the principal trace is a linear map and that the principal norm is multiplicative. For field extensions and central simple algebras, they coincide with the usual trace and norm resp. reduced trace and reduced norm.

Integrality and orders. Suppose $A$ is a finite-dimensional $K$-algebra with $K$ the field of fractions of some Noetherian and integrally closed ring $R$. You call an element $a \in A$ integral (over $R$) when the following equivalent conditions are satisfied:

• $a$ is the root of a monic polynomial in $R[X]$.
• The minimal polynomial of $a$ is in $R[X]$.
• The characteristic polynomial of $a$ is in $R[X]$.
• The principal polynomial of $a$ is in $R[X]$.
• The $R$-algebra generated by $a$ is finite (=finitely generated as an $R$-module).
• $a$ is contained in a finite $R$-algebra.

Define an order in $A$ to be a finite $R$-subalgebra that generates $A$ over $K$. When either

• $A$ is separable (think $A$ = field extension of $K$ with $\operatorname{char}(K) = 0$ or $A$ = quaternion algebra over $K$)
• $A$ is only semi-simple, but $K$ is a global field and $R$ is the localization of $\mathcal O_K$ at finitely many places (think $K = \mathbb F_q(T)$, $R = \mathcal O_K$, $A$ = field extension that can be inseparable)

one can show that an $R$-algebra $\mathcal O \subset A$ is an order iff it consists of integral elements and generates $A$ over $K$. (The implication $\implies$ is immediate, $\Longleftarrow$ is harder.)