# Norm for (finite) ring extensions #

Suppose we have an `R`

-algebra `S`

with a finite basis. For each `s : S`

,
the determinant of the linear map given by multiplying by `s`

gives information
about the roots of the minimal polynomial of `s`

over `R`

.

## Implementation notes #

Typically, the norm is defined specifically for finite field extensions. The current definition is as general as possible and the assumption that we have fields or that the extension is finite is added to the lemmas as needed.

We only define the norm for left multiplication (`Algebra.leftMulMatrix`

,
i.e. `LinearMap.mulLeft`

).
For now, the definitions assume `S`

is commutative, so the choice doesn't
matter anyway.

See also `Algebra.trace`

, which is defined similarly as the trace of
`Algebra.leftMulMatrix`

.

## References #

The norm of an element `s`

of an `R`

-algebra is the determinant of `(*) s`

.

## Equations

- Algebra.norm R = LinearMap.det.comp ↑(Algebra.lmul R S).toRingHom

## Instances For

If `x`

is in the base ring `K`

, then the norm is `x ^ [L : K]`

.

If `x`

is in the base field `K`

, then the norm is `x ^ [L : K]`

.

(If `L`

is not finite-dimensional over `K`

, then `norm = 1 = x ^ 0 = x ^ (finrank L K)`

.)