Axiom Of Choice

We have the following indirect implication of form equivalence classes:

72 \(\Rightarrow\) 72 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
72 \(\Rightarrow\) 72	Prime ideal theorems for Boolean algebras and the axiom of choice, Tarski, A. 1954b, Bull. Amer. Math. Soc.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
72:	Artin-Schreier Theorem: Every field in which \(-1\) is not the sum of squares can be ordered. (The ordering, \(\le \), must satisfy (a) \(a\le b\rightarrow a + c\le b + c\) for all \(c\) and (b) \(c\ge 0\) and \(a\le b\rightarrow a\cdot c\le b\cdot c\).)
72:	Artin-Schreier Theorem: Every field in which \(-1\) is not the sum of squares can be ordered. (The ordering, \(\le \), must satisfy (a) \(a\le b\rightarrow a + c\le b + c\) for all \(c\) and (b) \(c\ge 0\) and \(a\le b\rightarrow a\cdot c\le b\cdot c\).)

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