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Revelation
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 The Bottomless Pit The bottomless pit is mentioned in revelation in the description of the beast rising from the pit as well as in the sounding of the fifth trumpet. The algebra of the bottomless pit is simply stated as "algebraic closure". In defining a prime subfield, say GF(2) we may place any extension over it we wish, and we will always have a field of order a prime power of 2. We need not restrict ourselves to algebra modulo 2, but may construct fields of order a power of any prime number. Alebraic closure follows along the lines that there if there is an field extension of order p^n and another of order p^m then there is an extension of order p^(nm) that contains both those fields as subfields. We then use Zorn's lemma to simply "choose" the field at the very top of the chain: This field contains every possible finite field of characteristic p (we do arithmetic mod p). Then, as every extension field is a subfield of algebraic closure we have the consquence that it contains not just every finite field of characteristic p, but every extension of itself: We have a system that may not be extended further. It is safe to say that we may not attach a root of an irreducible to the field to extend it, as the closure already contains such a root. This makes sense of the statement that it contains every extension of itself: For there are no more "irreducibles" to find roots from, to attach them. However there is one thing we may surmise: that the algebraic closure is the union of every finite field of characteristic p. Also, since there is no root of order p in the fields in union, (there is no multiplicative element of order p) we may state that this lock on all the field's elements - the existence of a multiplicative root of order p in a field of characteristic p is such as to be correct for algebraic closure also. There is a theorem called the isomorphism extension theorem that will account for this too. Every isomorphism of a field of characteristic p is embedded in the algebraic closure. Were there to be a root of order p in any GF(p,n)* there would be a subfield extended by this root inside algebraic closure. Yet this is not possible since then there would be a non-trivial "multiplicative kernel" of the frobenius automorphisms of the field. We would "contradict" on algebra modulo p. So we see that without any loss of generality the bottomless pit is safely under lock and key; We may also surmise that God reveals that in this sense He is not the algebraic closure of the finite field of characteristic 2, or perhaps any other prime. However the pit in the text is clearly "opened", We do indeed join a root of multiplicative order two to the fields in view. For this, we require one of two things: either we change characteristic to some other prime or composite integer, or we attach an additive element of order p^n (n>1) to the field. We could unlock the whole system completely by allowing any order additive elements as well - so that we have an undefined characteristic: (being infinite. The characteristic is the smallest integer n such that nx = 0 for every element x. If n is left completely unbound then we have no field (and no multiplicative group.)) If we restrict ourselves to m "a prime" so that mx=0 for some m for each x, then we have zero divisors if the charactersitic is a product of primes and therefore there is an additive element of order pq say (p,q prime) Then there are elements of order p and q that divide 0. So then, adding or adjoining an element of order p^2 , leaves us with p elements of order p in F+. Under a homomorphism sending these p elements to zero, we may map onto a finite field of characteristic p. At the least this gives us a field as a subfield of our new ring, but also allows us to define an ideal over the field. (We may have a root of greater multiplicity.) The frobenius map would again be the map we require. However these p elements in the "kernel" of the map are not elements of a multiplicative group - if they as an ideal were to contain unity then the whole ring would be a field. We instead have a set of idempotents (or zero divisors). In any event, the field GF(2,n) would be a subfield and in this sense would not be "obscured". Rather, we shift characteristic altogether! In changing to another characteristic, allowing an element of order two in F* we omit any field GF(2,n) from ever being a subfield of our new structure. There is no subset of the prime subfield that would permit the elements of GF(8)+ or GF(4)+ to "exist". The "opening" of the bottomless pit is such as to obscure the seven cycles (air or spirit, C7) and the sun (the octal) We may safely assume that the only systems we use are fields with their additive subgroups under permutation with multiplication. Those extensions of characteristic two above the degree of GF(8) are also part of this system. However it is weight of numbers or the "swarm" of locusts that drive the pit - its purpose is to obscure the clearly defined God of heaven amongst crowds. GF(4) and GF(8) need not be a subfield of GF(2,n). Do not think of these structures as "spirits" or "Gods" but instead think of them as a lifeless body within which elements are made to come alive when they are played as parts on a stage. The animating principle - (for there is none) for these structures is the mentality of the crowd under delusion - they perceive some "life" but do not realise it is the "life" of the image made to the beast, and they are the generators for it. |
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