# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 4 Results
### Construction of Algebraic Rings via Nonstandard Algebraic Extensions of Galois Fields  
Let us make the following two linear combinations:  
\[e_{10}=q_{0}+i,e_{2}=q_{0}-i \tag{20}\]  
One can see that the elements \(e_{1,2}\) cancel each other out  
\[e_{10}e_{20}=(q_{0}+i)(q_{0}-i)=-1+1=0, \tag{21}\]  
since \(q_{0}\) and \(i\) are solutions of the same Equation (8). Consequently, we can claim that the object obtained using algebraic extensions of the proposed type is not a field since it has divisors of zero. Let us define the elements  
\[e_{1,2}=\alpha(q_{0}\pm i) \tag{22}\]  
where the multiplier \(\alpha\) is determined on the basis of the condition  
\[e_{1}+e_{2}=1 \tag{23}\]or  
\[2\alpha q_{0}=1 \tag{24}\]  
With this choice of multiplier \(\alpha\), elements \(e_{1,2}\) are idempotent. Indeed,  
\[e_{1,2}e_{1,2}=\alpha^{2}(q_{0}\pm i)(q_{0}\pm i)=2\alpha^{2}q_{0}(q_{0}\pm i)= \alpha(q_{0}\pm i) \tag{25}\]  
This result corresponds to one of the general theorems of the theory of algebraic ideals [30]: there is a kind of ring \(R\) that decomposes into a direct sum of ideals \(r_{k}\). \[R=r_{1}+r_{2}+\cdots+r_{n} \tag{26}\]  
Each of these ideals is generated by idempotent elements \(e_{i}\)  
\[r_{k}=Re_{k} \tag{27}\]  
which cancel each other, and their sum is equal to the unit of the ring \(R\). \[\sum_{k}e_{k}=1 \tag{28}\]  
For further calculations, it is convenient to represent idempotent elements (22) in the form  
\[e_{1,2}=\beta\pm\alpha i \tag{29}\]  
where  
\[e_{10}e_{20}=(q_{0}+i)(q_{0}-i)=-1+1=0, \tag{21}\]0  
\[e_{10}e_{20}=(q_{0}+i)(q_{0}-i)=-1+1=0, \tag{21}\]1  
The use of Formula (31) is legitimate since any nonzero element of a Galois field has an inverse. In particular, it can be shown by direct calculation that for Galois fields of the considered type, the element \(\beta\) can be found in the general case as  
\[e_{10}e_{20}=(q_{0}+i)(q_{0}-i)=-1+1=0, \tag{21}\]2  
Starting from Formula (29), we can immediately specify the matrix representation for idempotent elements \(e_{1,2}\)  
\[e_{1}=\beta q_{0}+\alpha i\ \leftrightarrow E_{1}= \begin{pmatrix}\beta&- \alpha\\ \alpha&\beta\end{pmatrix} =\beta E_{0}+\alpha I \tag{33}\]  
\[e_{2}=\beta q_{0}-\alpha i\ \leftrightarrow E_{2}= \begin{pmatrix}\beta& \alpha\\ -\alpha&\beta\end{pmatrix} =\beta E_{0}-\alpha I \tag{34}\]  
where all matrix elements belong to \(GF(p)\) and \(p-1\) is divisible by \(4\). As one would expect, the matrices (33) and (34) cancel each other out. \[ \begin{pmatrix}\beta&-\alpha\\ \alpha&\beta\end{pmatrix}\begin{pmatrix}\beta&\alpha\\ -\alpha&\beta\end{pmatrix} = \begin{pmatrix}\beta^{2}+\alpha^{2}&0\\ 0&\beta^{2}+\alpha^{2}\end{pmatrix} = \begin{pmatrix}0&0\\ 0&0\end{pmatrix}  \tag{35}\]In deriving Formula (35), it is taken into account that \(\beta=\alpha q_{0}\) and \(q_{0}^{2}=-1\) and, hence, \(\alpha^{2}=-\beta^{2}\). Similarly, it is proved that the matrices \(\mathbf{E}_{i}\) are idempotent since \(2\beta=1\). \[\mathbf{E}_{i}\mathbf{E}_{i}= \begin{pmatrix}\beta^{2}-\alpha^{2}&\mp 2\alpha\beta\\ \pm\alpha\beta&\beta^{2}-\alpha^{2}\end{pmatrix} = \begin{pmatrix}2\beta^{2}& \mp 2\alpha\beta\\ \pm 2\alpha\beta&2\beta^{2}\end{pmatrix} = \begin{pmatrix}\beta&\mp\alpha\\ \pm\alpha&\beta\end{pmatrix}  \tag{36}\]  
Thus, it can be seen that the classical method of algebraic expansions leads to the formation of fields, and the proposed method leads to the formation of algebraic rings. Let us give for clarity concrete examples of the construction of idempotent elements in Galois fields \(GF(17)\) and \(GF(13)\), which satisfy the criterion formulated above (number \(p-1\) is divisible by \(4\)). In the field \(GF(17)\), the element inverse to \(2\) is \(\beta=-8\), and the root of Equation (8) is the element \(q_{0}=4\). The element inverse to it is \(q_{0}^{-1}=-4\). Idempotent elements are represented in the form  
\[e_{1,2}=-8\pm 2i \tag{37}\]  
The fact can be proven by direct calculation. \[e_{1,2}e_{1,2}=(-8\pm 2i)^{2}=64-4\mp 32i=-8\pm 2i(17) \tag{38}\]  
These elements cancel each other out.