mathlibdocumentation

field_theory.cardinality

Cardinality of Fields #

In this file we show all the possible cardinalities of fields. All infinite cardinals can harbour a field structure, and so can all types with prime power cardinalities, and this is sharp.

Main statements #

• fintype.nonempty_field_iff: A fintype can be given a field structure iff its cardinality is a prime power.
• infinite.nonempty_field : Any infinite type can be endowed a field structure.
• field.nonempty_iff : There is a field structure on type iff its cardinality is a prime power.
theorem fintype.is_prime_pow_card_of_field {α : Type u_1} [fintype α] [field α] :

A finite field has prime power cardinality.

theorem fintype.nonempty_field_iff {α : Type u_1} [fintype α] :

A fintype can be given a field structure iff its cardinality is a prime power.

theorem fintype.not_is_field_of_card_not_prime_pow {α : Type u_1} [fintype α] [ring α] :
theorem infinite.nonempty_field {α : Type u} [infinite α] :

Any infinite type can be endowed a field structure.

theorem field.nonempty_iff {α : Type u} :

There is a field structure on type if and only if its cardinality is a prime power.