1: - Trinity In Christ

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Metamath Aftermath Introduction Loose Ends Nothing Less Than Zero Bungee Leap Messianism Three Bags Full Sir! Ultrafilters Introduction Ultrafilters, Insert Coin Application By Anselm Principal, Or Non-Principal Finite, Or Infinite? Algebra, Abstract Enough? Introduction Group Axioms Ring And Field Axioms Subrings, Subfields Modular Arithmetic Ideals Of Rings and Fields Morphisms / Factor Rings Galois Groups And Finite Fields Permutation Groups K4 Groups And The Octal The Finite Fields GF(4) and GF(8) Normal Subgroups Simple Groups A5 Is Simple A Proposal For The Trinity Introduction Modelling The Holy Trinity Of What Meaning Is It? Tested Whilst Surprised Three Bags Full – Of What? Ultrafilter In Strange Twist Limit Of Construction Introduction Divergence Holy Trinity – The Only Finite Pantheism – Most High God Lord Of Hosts – Angels Advanced Symmetry Introduction Operands vs. Operations 1: - Trinity In Christ 2: - Trinity Through The Father 3: - Trinity By The Holy Spirit They Do It With Mirrors Ultrafilters Of Trinity The Nature Of God Introduction Reconcilable God And The Dialectic Eucharistia A Child Called Alice The Certainty Of An Open Door And His Coming No System Of Works Alpha And Omega None Other Name Under Heaven Given Among Men The Comforter – Open Doors The 6, 7, 8 Cycle Introduction The 6, 7, 8 Cycle The Devil’s Jealousy Numbering The Beast The Four Horsemen The Seven Churches The Beast’s Deadly Wound The Vision Thereof Sure What A Tangled Web We Weave The Ten Kings – Ten Horns Of The Beast The Leaven Of Divorce – Even He Is The Eighth The False Prophet And The Image Of The Beast If Thy Right Hand Offend Thee The Mechanics Of Temptation Introduction Simple Transposition Factoring To A5 The Final Six The Pride Of Life The Ten Kings The Image To The Beast The Consequences Of The Image The Seventh Angel Sounds The Restrainer Have You Tried Turning It Off And On Again? Wise Virgins Introduction The Parable Of The Ten Virgins Left Handed And Right Handed – The Father’s Throne Sit Thou At My Right Hand Octal Reversal And Left/Right Equivalence The Sabbath Rest Of Christ – Spiritual Israel Is His Body The 144 Octal Reversals Seven Seals, Trumpets And Vials – Thunders? An Application Twelve Thrones Over Israel Introduction The Maker’s Dozen The Twelve Tribes A Matter Of Life And Death Life Beside The Right Hand But Whom Say Ye That I Am? Introduction Right And Left Hand Invariance Left And Right Hands As Vectors Scriptural Comparison Remaining Calculations Walking Amongst The Candlesticks Amalgam 666 Introduction For A Mark Between Thine Eyes 666 Outside Of Christ 660 or 666? - The Image With Life Death Shall Flee From Them Wormwood = Death Abaddon = Hell The Heart Of Mystery Babylon - A Cage For Every Bird Hell Follows After The Insanity Of Satan Introduction Satan's Left And Right Hand Conterfeits Discontinuity - A Deadly Wound The Madness Of The Dragon A Bag Full Of Holes Inept, To Claim The Least The Beast Which Thou Sawest No Other Name Under Heaven The Spirit Within Us Lusteth To Envy Two Horns Like A Lamb No Mass Culling 6, 7, 8 Cycle Revisited Introduction With Apologies To The Reader Rightly Dividing The Word Of Truth The Deadly Wound Healed New Beginnings Hot, Cold And Lukewarm The Collision In The Octal Regressive Or Regenerative? Seven Spirits More Wicked Maths From The Gospels Introduction Our Father, Which Art In Heaven Hallowed Be Thy Name Thy Kingdom Come Thy Will Be Done On Earth, As It Is In Heaven Give Us Today Our Daily Bread Forgive Us Our Sins... Lead Us Not Into Temptation... For Thine Is The Kingdom The Power And The Glory, Forever	1: - Trinity In Christ Let us make a start then. The Klein four-group has, as its name suggests, four elements. As a group it obeys the group axioms and it is also a group comprised of prime power cyclic factors of order 2, and is therefore isomotrphic to the additive group of the finite field of order two. We will consider the operations of the finite field upon the Klein four group. Here are the tables for the operations of addition "+" and multiplication, "*". (Multiplication in a finite field) Alternatively we may consider the elements as 0, 1, x and x+1 when addition is performed with coefficients modulo 2, (The remainder on dividing by two - why there is "only 0 and 1") and multiplication likewise with the extra condition that we take the remainder when dividing any products by an irreducible polynomial with coefficient modulo two. x² + x + 1 is such a polynomial. (Try putting the values 0 and 1 as the values of x, and not that any root of this polynomial could not be in The field of two elements only, our "0 and 1". - The equation does not equal zero for either value.) The important thing to note is that the multiplicative group is also cyclic, of order three. Multiplication appears to be similar to addition, but the exponents are computed "modulo 3" here. This is true in general for fields - For every prime power there is a unique (up to isomorphism) finite field of that order and the multiplicative group is cyclic. Notice also that the additive group is composed of three elements - any of these could be our unity or "1". Any element is the product of the other two. We will simply state that as far as unity in the multiplication table is concerned: it is the product of the other two elements, Zero, we will not consider in this: although we will approach this element later. Whichever non-zero member of the additive group we choose then, we will set as our "1" and make it both the product additively and multiplicatively of the other two. We therefore have three possible choices for unity, and only when we set a particular element as unity can we do this: There is an isomorphism between the two remaining states where the other two elements may switch in position. This automorphism can be generated multiplicatively by squaring each element before or after performing addition or multiplication. A binomial expansion will easily show that coefficients of lesser powers in the product are 0 modulo 2, and we call this automorphism the Frobenius automorphism. It is a general state of affairs that all automorphisms of finite fields are obtained by repeated application of the Frobenius mapping, and to the power of its prime characteristic, and there are in number an equal number of possible automorphisms (including self to self) as the degree of the field extension. (4 is two squared, hence two automorphisms. A field of order 27 would be of degree three over its prime subfield, and there would be three such automorphisms.) It is true in general that Every automorphism leaves a subfield fixed. Here it is the prime subfield, but there is also an automorphism of the field of 81 elements leaving a subfield of 9 elements fixed, as well as the prime subfield of three elements. However, this is only possible if there actually is a subfield of that order. The degree of the subfield over the prime subfield must divide the degree of the whole field over the prime subfield. Hence there is a subfield of order 8 in a field of order 64, but not a subfield of order four in a field of order eight, (Even though there is such a subfield in a field of order 64.) So then, we consider our elements as members of the Klein four group. It would appear that there is an equivocation between the operations of addition and multiplication and the remaining two elements. This is not the case as we will see as we progress, rather the statement is one of "there exists a such and such". It is enough to see the symmetry that either additively or multiplicatively every unital choice is the product of the other two non-zero elements. We will move onto the case of the Octal group in the next page. Continue To Next Page Return To Section Start Return To Previous Page