ordered-field

# ordered-field - There is one order-complete ordered-field...

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Unformatted text preview: There is one order-complete ordered-field Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 15 September, 2010 (at 10:57 ) § Contents Rings . . . . . . . . . . . . . . . . . . . . . . . 1 Totally-Ordered Sets . . . . . . . . . . . . . . . 1 Least upper-bound property [ LUBP ] . . . 1 LUBP theorem . . . . . . . . . . . . . . . . . . 1 Making a Real Assumption . . . . . . . . . 2 Making a Real Assumption . . . . . . . 2 Ordered-fields . . . . . . . . . . . . . . . . . . . . . . . 2 Ordered-field lemma . . . . . . . . . . . . . . . . 2 Archimedean fields . . . . . . . . . . . . . . . . . . . . 3 Archy lemma . . . . . . . . . . . . . . . . . . . 3 OC ⇒ Archimedean theorem . . . . . . . . . . . 3 Order-dense lemma . . . . . . . . . . . . . . . . 3 Complete ordered-field(s) . . . . . . . . . . . . . . . . 4 Rings. In a ring (Γ , + , Γ , · , 1 Γ ), if there is a posint n so that 1 Γ + 1 Γ + n ... + 1 Γ equals Γ , then the smallest such n is called the characteristic of the ring, and I write Char(Γ) = n . If no such posint exists, then I will write Char(Γ) = ∞ ; however , the standard term is Char(Γ) = 0, and you will see this in algebra texts and in some of my notes. A ring is commutative ( abbrev., comm-ring ) if its multiplication is commutative. In a comm-ring Γ, a zero-divisor α ∈ Γ admits a non-zero elt β ∈ Γ ( this β need not be unique ) so that αβ = Γ . Use ZD for zero-divisor . [ Letting ≡ denote ≡ 12 , in the Z 12 ring, 9 is a ZD, since 9 · 8 ≡ 0, yet 8 6≡ 0. OTOHand, even though 5 · 24 ≡ 0, this doesn’t show that 5 is a Z 12 –ZD, since 24 ≡ 0. ] An integral domain Γ is a commutative ring with no ZDs except for Γ , the trivial ZD . If the charac- teristic of an integral domain is finite, then Char(Γ) is a prime number. In particular, this holds if Γ is a field. 1 : Fact. If Γ is a field of finite order ( finite cardinality ) then | Γ | = p k for some prime p and posint k . Con- versely, for each such prime p and k ∈ Z + , there exists a field of order p k , and this field is unique upto field- isomorphism. ♦ Partial proof. For the prime p := Char(Γ), there is a copy of Z p inside Γ, making Γ a Z p-vectorspace. Letting k be the dimension of this vectorspace, then, we obtain | Γ | = p k . The remaining Fact s take a fair amount of work to prove. Totally-Ordered Sets. A TOS (Γ , ≺ ) has an antireflexive, transitive relation ≺ so that for each α 6 = β in Γ, either α ≺ β or α β ....
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## This note was uploaded on 01/26/2012 for the course MAC 3472 taught by Professor Jury during the Fall '07 term at University of Florida.

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