ring_theory.norm

# 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.left_mul_matrix, i.e. algebra.lmul_left). 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.left_mul_matrix.

## References #

noncomputable def algebra.norm (R : Type u_1) {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] :
S →* R

The norm of an element s of an R-algebra is the determinant of (*) s.

Equations
theorem algebra.norm_apply (R : Type u_1) {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] (x : S) :
theorem algebra.norm_eq_one_of_not_exists_basis (R : Type u_1) {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] (h : ¬∃ (s : finset S), nonempty (basis s R S)) (x : S) :
theorem algebra.norm_eq_matrix_det {R : Type u_1} {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] {ι : Type w} [fintype ι] [decidable_eq ι] (b : R S) (s : S) :
theorem algebra.norm_algebra_map_of_basis {R : Type u_1} {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] {ι : Type w} [fintype ι] (b : R S) (x : R) :
(algebra.norm R) ( S) x) =

If x is in the base field K, then the norm is x ^ [L : K].

@[simp]
theorem algebra.norm_algebra_map {K : Type u_4} {L : Type u_5} [field K] [field L] [ L] (x : K) :
(algebra.norm K) ( L) x) =

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).)

theorem algebra.power_basis.norm_gen_eq_coeff_zero_minpoly {S : Type u_2} [comm_ring S] {K : Type u_4} [field K] [ S] (pb : S) :
(algebra.norm K) pb.gen = ((-1) ^ pb.dim) * (minpoly K pb.gen).coeff 0

Given pb : power_basis K S, then the norm of pb.gen is (-1) ^ pb.dim * coeff (minpoly K pb.gen) 0.

theorem algebra.power_basis.norm_gen_eq_prod_roots {S : Type u_2} [comm_ring S] {K : Type u_4} {F : Type u_6} [field K] [field F] [ F] [ S] (pb : S) (hf : (minpoly K pb.gen)) :

Given pb : power_basis K S, then the norm of pb.gen is ((minpoly K pb.gen).map (algebra_map K F)).roots.prod.

theorem algebra.norm_eq_zero_iff_of_basis {R : Type u_1} {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] {ι : Type w} [fintype ι] [is_domain S] (b : R S) {x : S} :
(algebra.norm R) x = 0 x = 0
theorem algebra.norm_ne_zero_iff_of_basis {R : Type u_1} {S : Type u_2} [comm_ring R] [is_domain R] [comm_ring S] [ S] {ι : Type w} [fintype ι] [is_domain S] (b : R S) {x : S} :
@[simp]
theorem algebra.norm_eq_zero_iff {K : Type u_4} {L : Type u_5} [field K] [field L] [ L] [ L] {x : L} :
(algebra.norm K) x = 0 x = 0

See also algebra.norm_eq_zero_iff' if you already have rewritten with algebra.norm_apply.

@[simp]
theorem algebra.norm_eq_zero_iff' {K : Type u_4} {L : Type u_5} [field K] [field L] [ L] [ L] {x : L} :
linear_map.det ( L) x) = 0 x = 0

This is algebra.norm_eq_zero_iff composed with algebra.norm_apply.

theorem algebra.norm_eq_norm_adjoin (K : Type u_4) {L : Type u_5} [field K] [field L] [ L] [ L] [ L] (x : L) :
theorem intermediate_field.adjoin_simple.norm_gen_eq_one {K : Type u_4} {L : Type u_5} [field K] [field L] [ L] {x : L} (hx : ¬ x) :
theorem intermediate_field.adjoin_simple.norm_gen_eq_prod_roots {K : Type u_4} {L : Type u_5} {F : Type u_6} [field K] [field L] [field F] [ L] [ F] (x : L) (hf : (minpoly K x)) :
theorem algebra.norm_eq_prod_embeddings_gen {K : Type u_4} {L : Type u_5} [field K] [field L] [ L] (E : Type u_7) [field E] [ E] (pb : L) (hE : (minpoly K pb.gen)) (hfx : (minpoly K pb.gen).separable) :
E) ((algebra.norm K) pb.gen) = ∏ (σ : L →ₐ[K] E), σ pb.gen
theorem algebra.norm_eq_prod_roots {K : Type u_4} {L : Type u_5} {F : Type u_6} [field K] [field L] [field F] [ L] [ F] [ L] [ L] {x : L} (hF : (minpoly K x)) :
theorem algebra.prod_embeddings_eq_finrank_pow {K : Type u_4} {L : Type u_5} (F : Type u_6) [field K] [field L] [field F] [ L] [ F] (E : Type u_7) [field E] [ E] [ F] [ F] [ F] [ F] (pb : L) :
∏ (σ : F →ₐ[K] E), σ ( F) pb.gen) = (∏ (σ : L →ₐ[K] E), σ pb.gen) ^
theorem algebra.norm_eq_prod_embeddings (K : Type u_4) {L : Type u_5} [field K] [field L] [ L] (E : Type u_7) [field E] [ E] [ L] [ L] {x : L} :
E) ((algebra.norm K) x) = ∏ (σ : L →ₐ[K] E), σ x

For L/K a finite separable extension of fields and E an algebraically closed extension of K, the norm (down to K) of an element x of L is equal to the product of the images of x over all the K-embeddings σ of L into E.

theorem algebra.is_integral_norm {S : Type u_2} [comm_ring S] (K : Type u_4) {L : Type u_5} [field K] [field L] [ L] [ L] [ K] [ L] [ L] [ L] {x : L} (hx : x) :