# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 4 Results
### Representation of Operators Through Algebraic Extensions  
In this paper, we consider Galois fields in which Equation (8) has a solution. This, however, does not exclude that the classical method of algebraic expansions cannot be applied to them. For this purpose, one can use, in particular, any equation of the same form but which has no solution. \[x^{2}+g=0 \tag{43}\]  
The simultaneous use of the extensions generated by Equations (8) and (43) makes it possible to ensure that an arbitrary operator describing a coordinate transformation is represented in a form similar to (1) but containing two kinds of adjoint elements. By attaching to the basic field the root \(\theta\) of Equation (43), we can pass to the elements expressed by Formula (1). Passing again to the matrix representation in accordance with Formula (15), we can see that an arbitrary element \(A\) obtained by using the element \(\theta\) can be put in correspondence to the next matrix. \[A\leftrightarrow A= \begin{pmatrix}A_{1}&-gA_{2}\\ A_{2}&A_{1}\end{pmatrix} =A_{1} \begin{pmatrix}1&0\\ 0&1\end{pmatrix} +A_{2} \begin{pmatrix}0&-g\\ 1&0\end{pmatrix} =A_{1}\textbf{E}_{0}+A_{2}\textbf{G} \tag{44}\]  
Elements \(\theta\) and \(i\) are different, so it is acceptable to consider the combination  
\[w=(A_{1}+A_{2}\theta)(\beta+\alpha i)+(B_{1}+B_{2}\theta)(\beta-\alpha i) \tag{45}\]  
This combination satisfies Formulas (26) and (27) under the assumption that algebraic ideals \(r_{1,2}\) correspond to classical algebraic extensions of the basic Galois field. Indeed, as follows from Formula (29), expression (45) can be rewritten in the form  
\[w=(A_{1}+A_{2}\theta)e_{1}+(B_{1}+B_{2}\theta)e_{2} \tag{46}\]  
The correctness of the representations (45) and (46) can be demonstrated by going to the matrix representation again. We have  
\[\textbf{W}= \begin{pmatrix}w_{11}&w_{12}\\ w_{21}&w_{22}\end{pmatrix} =(A_{1}\textbf{E}_{0}+A_{2}\textbf{G})(\beta \textbf{E}_{0}+\alpha I)+(B_{1}\textbf{E}_{0}+B_{2}\textbf{G})(\beta\textbf{ E}_{0}-\alpha\textbf{I}) \tag{47}\]  
We will show that any matrix \(W\) given over a Galois field of the considered type admits representation (47). This will prove that the ring of such matrices is isomorphic to an algebraic object, which is formed by simultaneous use of both classical and nonstandard methods of algebraic extensions. Opening the brackets in Formula (47), we obtain  
\[\textbf{W}=\beta((A_{1}+B_{1})\textbf{E}_{0}+(A_{2}+B_{2})\textbf{G})+\alpha ((A_{1}-B_{1})\textbf{I}+(A_{2}-B_{2})\textbf{G}\textbf{I}) \tag{48}\]  
We will use the following variables  
\[x_{1}=\beta(A_{1}+B_{1}),\ x_{2}=\beta(A_{2}+B_{2}),\ x_{3}=\alpha(A_{1}-B_{1 }),\ x_{4}=\alpha(A_{2}-B_{2}) \tag{49}\]  
\[\text{Then }\textbf{W}=(x_{1}\textbf{E}_{0}+x_{2}\textbf{G})+(x_{3}\textbf{I}+x _{4}\textbf{G}\textbf{I}) \tag{50}\]  
Formula (50) is equivalent to the following system of linear equations:  
\[ \begin{cases}x_{1}-gx_{4}=w_{11}\\ x_{1}-x_{4}=w_{22}\end{cases}  \tag{51}\]  
\[ \begin{cases}-gx_{2}-x_{3}=w_{12}\\ x_{2}+x_{3}=w_{21}\end{cases}  \tag{52}\]  
It is not difficult to specify their explicit solutions. \[x_{4}=(g-1)^{-1}(w_{22}-w_{11}) \tag{53}\]  
\[x_{1}=(g-1)^{-1}(gw_{22}-w_{11}) \tag{54}\]  
\[x_{2}=-(g-1)^{-1}(w_{12}+w_{21}) \tag{55}\]  
\[x_{3}=(g-1)^{-1}(w_{12}+gw_{21}) \tag{56}\]These solutions are elementary, but they clearly show that the simultaneous use of two different methods of algebraic extensions allows one to pass from the basic Galois field to an algebraic ring isomorphic to the set of linear operators acting on vectors, corresponding to discrete coordinates in the plane. In particular, instead of the matrix representation of the operator describing the coordinate transformation, we can use its representation in the form of  
\[w=(x_{1}+x_{2}\theta)+(x_{3}i+x_{4}\theta i) \tag{57}\]  
The product of two matrices corresponding to coordinate transformations can also be reduced to a representation of the form (57).