- #1

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

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## Main Question or Discussion Point

I was wondering: what is the proof idea behind results such as:

(Every vector space has a basis) iff AoC

(All bases for a vector space have the same cardinality) iff AoC

(Every field has an algebraic closure) iff AoC

One direction is obvious, but I have no idea how to begin the other!

As a related question, what is the status of (all algebraic closures of a given field are isomorphic), relative to the AoC?

(Every vector space has a basis) iff AoC

(All bases for a vector space have the same cardinality) iff AoC

(Every field has an algebraic closure) iff AoC

One direction is obvious, but I have no idea how to begin the other!

As a related question, what is the status of (all algebraic closures of a given field are isomorphic), relative to the AoC?