## Infinite primes and ordered fields |

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algebraic extension algebraic over Q algebraically closed field approaches-to-a Archimedean conic prime Archimedean modes Archimedean order ArchP belongs to F bijection closed under addition closure coefficients complex place cone in F containing P properly continuous functions Corollary defined division prime exists extension of Q F is algebraic field F field of quotients finite formally real field full Archimedean AC full Archimedean prime Hausdorff space Hence homomorphism hypothesis imbedding implies infinite prime integral intersection irreducible polynomial Let F Let G Let z belong maximal ideal member of F modal prime mode in F nonempty nonzero member open set order in F orderable field orders of F place of F positive rational preprime primacy prime in F primitive Proof Proposition proved real functions real place S(z+ shows subring subset Suppose transcendental universally positive elements universally positive number valuation ring weak topology x-1 belongs zero