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feat(field_theory/tower): tower law (#3355)
Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
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/- | ||
Copyright (c) 2020 Kenny Lau. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Kenny Lau | ||
-/ | ||
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import ring_theory.algebra_tower | ||
import linear_algebra.finite_dimensional | ||
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/-! | ||
# Tower of field extensions | ||
In this file we prove the tower law for arbitrary extensions and finite extensions. | ||
Suppose `L` is a field extension of `K` and `K` is a field extension of `F`. | ||
Then `[L:F] = [L:K] [K:F]` where `[E₁:E₂]` means the `E₂`-dimension of `E₁`. | ||
In fact we generalize it to algebras, where `L` is not necessarily a field, but just a `K`-algebra. | ||
## Implementation notes | ||
We prove two versions, since there are two notions of dimensions: `vector_space.dim` which gives | ||
the dimension of an arbitrary vector space as a cardinal, and `finite_dimensional.findim` which | ||
gives the dimension of a finitely-dimensional vector space as a natural number. | ||
## Tags | ||
tower law | ||
-/ | ||
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universes u v w u₁ v₁ w₁ | ||
open_locale classical big_operators | ||
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section field | ||
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open cardinal | ||
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variables (F : Type u) (K : Type v) (A : Type w) | ||
variables [field F] [field K] [ring A] | ||
variables [algebra F K] [algebra K A] [algebra F A] [is_algebra_tower F K A] | ||
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/-- Tower law: if `A` is a `K`-algebra and `K` is a field extension of `F` then | ||
`dim_F(A) = dim_F(K) * dim_K(A)`. -/ | ||
theorem dim_mul_dim' : | ||
(cardinal.lift.{v w} (vector_space.dim F K) * | ||
cardinal.lift.{w v} (vector_space.dim K A) : cardinal.{max w v}) = | ||
cardinal.lift.{w v} (vector_space.dim F A) := | ||
let ⟨b, hb⟩ := exists_is_basis F K, ⟨c, hc⟩ := exists_is_basis K A in | ||
by rw [← (vector_space.dim F K).lift_id, ← hb.mk_eq_dim, | ||
← (vector_space.dim K A).lift_id, ← hc.mk_eq_dim, | ||
← lift_umax.{w v}, ← (hb.smul hc).mk_eq_dim, mk_prod, lift_mul, | ||
lift_lift, lift_lift, lift_lift, lift_lift, lift_umax] | ||
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/-- Tower law: if `A` is a `K`-algebra and `K` is a field extension of `F` then | ||
`dim_F(A) = dim_F(K) * dim_K(A)`. -/ | ||
theorem dim_mul_dim (F : Type u) (K A : Type v) [field F] [field K] [ring A] | ||
[algebra F K] [algebra K A] [algebra F A] [is_algebra_tower F K A] : | ||
vector_space.dim F K * vector_space.dim K A = vector_space.dim F A := | ||
by convert dim_mul_dim' F K A; rw lift_id | ||
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namespace finite_dimensional | ||
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theorem trans [finite_dimensional F K] [finite_dimensional K A] : finite_dimensional F A := | ||
let ⟨b, hb⟩ := finite_dimensional.exists_is_basis_finset F K in | ||
let ⟨c, hc⟩ := finite_dimensional.exists_is_basis_finset K A in | ||
finite_dimensional.of_finite_basis $ hb.smul hc | ||
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/-- Tower law: if `A` is a `K`-algebra and `K` is a field extension of `F` then | ||
`dim_F(A) = dim_F(K) * dim_K(A)`. -/ | ||
theorem findim_mul_findim [finite_dimensional F K] [finite_dimensional K A] : | ||
findim F K * findim K A = findim F A := | ||
let ⟨b, hb⟩ := finite_dimensional.exists_is_basis_finset F K in | ||
let ⟨c, hc⟩ := finite_dimensional.exists_is_basis_finset K A in | ||
by rw [findim_eq_card_basis hb, findim_eq_card_basis hc, | ||
findim_eq_card_basis (hb.smul hc), fintype.card_prod] | ||
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end finite_dimensional | ||
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end field |
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