Finite fields are necessary for this study of the Revelation. Whilst it is fairly easy to construct a finite field, it is harder to come by the ingredients than it is to follow the recipe! In this section I include some of the more advanced mathematics not included in the book because unless you were to buy an algebra text to support the study, you will find yourself partly limited in the mathematical tools I had used throughout the book.

Constructing Finite Fields
The underlying conditions for constructing a finite field are given in this page. These proofs were taken from "A First Course In Abstract Algebra - Sixth Edition" (Fraleigh, J. B. :Addison Wesley Longman July 2000) I hope you will find the subjects depth and intricacy as beautiful as I.

For Any Prime Power
This page really shows the simplicity and elegance of the separable nature of finite fields. Every extension is built with the full machinery of the last page as if it was almost natural - one could almost not notice it! The relationships between the uniqueness of subfields and the degree of extension over the base field can not seem independent of all the prior machinery, yet all seems so flawlessly structured that there is no lesser word than "perfect" for these structures. These proofs are again taken from the same textbook as above.

The Multiplicative Group Is Cyclic
Again I briefly state the remainder of the argument of the math; the (almost) trivial addition to what went before, that complete description of the structure of the multiplicative part of every finite field. Almost a bit too brief a page for such a statement that is effective on every prime power over every degree of field, with all their subfields too etc! Such a small observation with huge consequence. I then include an outline of Rabin's irreducibility test for polynomials over finite fields, which really illustrates the connection between the irreducibles over a base field and the characteristic polynomial that shows that every multiplicative group of every finite extension of any finite field is also cyclic.

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