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Mathlib.FieldTheory.Tower

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 rings and modules, where L is not necessarily a field, but just a free module over K.

Implementation notes #

We prove two versions, since there are two notions of dimensions: Module.rank which gives the dimension of an arbitrary vector space as a cardinal, and FiniteDimensional.finrank which gives the dimension of a finite-dimensional vector space as a natural number.

Tags #

tower law

Tower law: if A is a K-module and K is an extension of F then $\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.

theorem rank_mul_rank (F : Type u) (K : Type v) (A : Type v) [CommRing F] [Ring K] [AddCommGroup A] [Algebra F K] [Module K A] [Module F A] [IsScalarTower F K A] [StrongRankCondition F] [StrongRankCondition K] [Module.Free F K] [Module.Free K A] :

Tower law: if A is a K-module and K is an extension of F then $\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.

This is a simpler version of lift_rank_mul_lift_rank with K and A in the same universe.

Tower law: if A is a K-module and K is an extension of F then $\operatorname{rank}_F(A) = \operatorname{rank}_F(K) * \operatorname{rank}_K(A)$.

theorem FiniteDimensional.left (F : Type u) [Field F] (K : Type u_1) (L : Type u_2) [Field K] [Algebra F K] [Ring L] [Nontrivial L] [Algebra F L] [Algebra K L] [IsScalarTower F K L] [FiniteDimensional F L] :

In a tower of field extensions L / K / F, if L / F is finite, so is K / F.

(In fact, it suffices that L is a nontrivial ring.)

Note this cannot be an instance as Lean cannot infer L.

theorem FiniteDimensional.right (F : Type u) (K : Type v) (A : Type w) [Field F] [DivisionRing K] [AddCommGroup A] [Algebra F K] [Module K A] [Module F A] [IsScalarTower F K A] [hf : FiniteDimensional F A] :

Tower law: if A is a K-vector space and K is a field extension of F then dim_F(A) = dim_F(K) * dim_K(A).

This is FiniteDimensional.finrank_mul_finrank' with one fewer finiteness assumption.