  Given commutative, unital rings $A$ and $B$ with a ring homomorphism $A\to B$
making $B$ free of finite rank as an $A$-module, we can ask for a "trace" or
"norm" homomorphism taking algebraic data over $B$ to algebraic data over $A$.
In this paper we we construct a norm functor for the data of a quadratic
algebra: given a locally-free rank-$2$ $B$-algebra $D$, we produce a
locally-free rank-$2$ $A$-algebra $\mathrm{Nm}_{B/A}(D)$ in a way that is
compatible with other norm functors and which extends a known construction for
\'etale quadratic algebras. We also conjecture a relationship between
discriminant algebras and this new norm functor.
