Abstract Algebra/Alphabet field

Prerequisites (recommended): Abstract Algebra/Fields Abstract Algebra/Rings, ideals, ring homomorphisms thumb|right|200px|The letters of the English alphabet correspond to places in a 3x3x3 cube where one position is occupied by zero or space. The study of algebraic number theory shows that there is a finite field for every power of a prime number. Therefore, the English alphabet, supplemented by a space, is a set of twenty- seven elements, three to the third power, so there is an algebraic field with elements in bijective correspondence with it. One needs the field with three elements, , to construct the larger field. Note that in one has modular arithmetic for which three is zero, four is one, and two is minus one. The field of twenty-seven elements has a group of units of twenty-six elements. Constructing the field amounts to generating the group as a cyclic group. Here another modulus arithmetic is used in a residue class ring. Start with the infinite-dimensional ring of polynomials and consider the ideal in it generated by the polynomial Polynomial f represents the class of zero in the ring of residues. f(X) = 0 means The cyclic group is generated by powers of a primitive root of f(X) = 0, and, whenever the cube of X appears, it is replaced with X + 2. If α is a primitive root of f, then is the group of units of the field. Recall The twenty-six trinomials in α correspond to the alphabet as follows: These 3x3 squares correspond to polynomials with respectively. A=1, B=α, C=α^2, and D=α+2. The three squares may be assembled into a 3x3x3 cube, representing the three-dimensional vector space over the field with three elements. This vector space reveals the additive structure of the alphabet field: addition is component-wise and modulo three. In a field, every nonzero element has a multiplicative inverse. Here A and N are their own inverses. Otherwise, note for 0 < j < 26. The following pairs are multiplicative inverses of each other in the alphabet field: BZ CY DX EW FV GU HT IS JR KQ LP MO. For example, MO =