  For a given support $L \in \mathbb{F}_{q^m}^n$ and a polynomial $g\in
\mathbb{F}_{q^m}[x]$ with no roots in $\mathbb{F}_{q^m}$, we prove equality
between the $q$-ary Goppa codes $\Gamma_q(L,N(g)) = \Gamma_q(L,N(g)/g)$ where
$N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In
particular, for $m=2$, that is, for a quadratic extension, we get
$\Gamma_q(L,g^q) = \Gamma_q(L,g^{q+1})$. If $g$ has roots in
$\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that
the difference of the dimensions of the two codes is bounded above by the
number of distinct roots of $g$ in $\mathbb{F}_{q^m}$. These identities provide
numerous code equivalences and improved designed parameters for some families
of classical Goppa codes.
