Residue-ring forms of finite bilinear modules #
On a finite bilinear module killed by n, the pairing takes values in the n-torsion of
ℚ/ℤ. Reading those values through ZMod.toRatAddCircle gives a symmetric bilinear form
with values in ZMod n. For prime n this lets finite-field linear algebra classify the
original pairing. The scalar action is any ZMod n-module structure on the given group;
additive maps are automatically linear for this action.
zmodForm is characterized by toRatAddCircle_zmodForm. Nondegeneracy, alternation and
isometry are preserved and reflected by this passage to the residue ring.
The pairing of a module killed by n, read as a ZMod n-valued bilinear form.
Instances For
Reading a residue-ring pairing back in ℚ/ℤ recovers the original pairing.
The residue-ring form is symmetric.
Nondegeneracy of the residue-ring form is equivalent to perfection of the original finite pairing.
Isometry of the finite pairings is equivalent to isometry of their residue-ring forms.