# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 4 Results  
### Nonstandard Method of Algebraic Extensions  
This method uses the following lemma, proved for the case when the field \(GF(p)\) corresponds to an integer \(p\), such that \(p-1\) is divisible by 4. We emphasize that from the point of view of the practical applications indicated above, this limitation is not so significant since, when mapping the terrain, the discretization character can be chosen based on considerations of convenience. Lemma 1: _If the number \(p-1\) is divisible by 4, then the equation_  
\[x^{2}+1=0 \tag{8}\]  
_has two solutions in the field \(GF(p)\)._Proof. : Let us represent the elements of the Galois field \(GF(p)\) as the following set:  
\[GF(p)\leftrightarrow\left\{-\frac{p-1}{2},-\frac{p-1}{2}+1,\ldots,-1,0,1,\ldots, \frac{p-1}{2}\right\} \tag{9}\]  
This representation is adequate because, except for the case of \(p=2\), which is not of interest, the number \(p\) is odd. This representation, among other things, makes the use of the minus sign in all subsequent calculations completely correct. In the most general case, all nonzero elements of the field \(GF(p)\) satisfy the equation  
\[x^{p-1}-1=0 \tag{10}\]  
By assumption, \(p-1\) is divisible by \(4\). Consequently, Equation (10) can be rewritten in the form  
\[\left(x^{4}\right)^{\frac{p-1}{4}}-1=0 \tag{11}\]  
Or  
\[\left(y\right)^{\frac{p-1}{4}}-1=0 \tag{12}\]  
where \(y=x^{4}\). The number of roots of Equation (12) cannot exceed \(\frac{p-1}{4}\), i.e., the elements satisfying Equation (12) cannot exhaust the considered Galois field. The remaining elements can therefore be considered as fourth roots of elements \(y\) satisfying Equation (12). Element \(1\) is obviously a particular solution of Equation (12). The fourth roots of \(1\) must therefore satisfy Equation (10), i.e., correspond to some elements of the original Galois field. All fourth roots of \(1\) can be obtained by extracting the square root of \(1\) (this will give two elements of the field \(1\) and \(-1\)) and then extracting the square roots of both of these elements. Consequently, the field under consideration includes two square roots of \(-1\). Consequently, Equation (8) has two solutions in the fields under consideration. The lemma is proven. \(\square\)  
For example, for the field \(GF(13)\), such solutions are \(x=\pm 5\), Formula (7). The basis for using the nonstandard method of algebraic extensions is the following theorem. Theorem 1.1: _A nonstandard algebraic extension of the field \(GF(p)\) that satisfies the conditions of the lemma proved above can be constructed by adjoining an algebraic element \(i\), which is formally a solution of Equation (8) but does not coincide with the solutions \(q_{0}\) of this equation in the specified field._  
Recall that the Galois fields \(GF(p^{n}),\ n\
eq 1\) are formed by the method of algebraic extensions. It consists in attaching one or more additional elements to the original field. An example of the use of this approach is the joining of an imaginary unit to a set of real numbers. Typically, in algebra, we use elements that are roots of equations that have no solutions in the original field. This theorem shows that this condition is not necessary. Proof. : The element \(i\) under consideration satisfies the equation  
\[i^{2}+1=0 \tag{13}\]  
but does not coincide with the solutions of Equation (8) in the main field, for which \(q_{0}^{2}+1=0\) also takes place. In the following, we will call the element \(i\) a logical imaginary unit of the second kind. Let us consider matrix representation for algebraic expansions of the proposed type. Using a formula similar to Formula (5), it can be seen that the components of the product of \(C_{1}+iC_{2}\) of two elements \(A_{1}+iA_{2}\) and \(B_{1}+iB_{2}\) are represented in the form  
\[C_{1}=A_{1}B_{1}-A_{2}B_{2};C_{2}=A_{1}B_{2}+A_{2}B_{1} \tag{14}\]  
or in matrix form  
\[ \begin{pmatrix}C_{1}\\ C_{2}\end{pmatrix} = \begin{pmatrix}A_{1}&-A_{2}\\ A_{2}&A_{1}\end{pmatrix}\begin{pmatrix}B_{1}\\ B_{2}\end{pmatrix}