# Electronic Fourier-Galois Spectrum Analyzer for the Field \(Gf(31)\)
## 3 Results
### Simplifying the Representation of Basis Functions
With this choice of integers \(\alpha_{i}\), the next formula also takes place:  
\[e_{1}+e_{2}+\cdots+e_{N}\equiv 1(P) \tag{17}\]  
In the considered case, an arbitrary element of the ring \(R\) represents it in the form:  
\[u=e_{1}u_{1}+e_{2}u_{2}+\cdots+e_{N}u_{N} \tag{18}\]  
where \(e_{i}\)--idempotent mutually canceling elements; \(u_{i}=0,1,2,\ldots,p_{i}-1\). The convenience of the representation (18) in calculations is that the elements of the \(u_{i}\) are multiplied independently, and the multiplication is performed modulo \(p_{i}\). Indeed, since \(e_{i}\) are mutually canceling idempotent elements, the product of two numbers \(u^{(1)}\) in \(u^{(2)}\) is expressed as follows:  
\[\theta^{p-1}=1 \tag{10}\]0  
Let us demonstrate the convenience of the representation (18) on a concrete example of the field used in this report \(GF(31)\). The number \(30\), equal to the number of non-zero elements in the given field, is the product of the three prime numbers \(2\), \(3\), and \(5\). Let us consider the ring of residue classes modulo \(30\). The idempotent mutually canceling elements are the numbers \(3\times 5=15\), \(3\times 2=6\), \(2\times 5=10\), i.e. :  
\[\theta^{p-1}=1 \tag{10}\]1  
By direct calculations, it can also be verified that for the case under consideration, all the \(\alpha_{i}=1\). Accordingly, when performing calculations modulo \(30\), Formula (18) takes the following form:  
\[\theta^{p-1}=1 \tag{10}\]2  
where:  
\[\theta^{p-1}=1 \tag{10}\]3  
that is, the ideals into which the ring under consideration decomposes can be put in correspondence to the fields \(GF(2)\), \(GF(3)\), and \(GF(5)\), respectively. Consequently, all non-zero elements of the field \(GF(31)\) are represented in the following form:  
\[\theta^{p-1}=1 \tag{10}\]4  
Indeed, if \(\theta\) is a primitive element, then for each element x belonging to the field under consideration, there exists an integer \(u\leq p-1\) such that \(x=\theta^{u}\). Representing u through Formula (21), we obtain Formula (23). Substituting the notations \(g_{1}=\theta^{15}\), \(g_{2}=\theta^{10}\), and \(g_{3}=\theta^{6}\), we obtain the following expression for an arbitrary element of the conserved field:  
\[\theta^{p-1}=1 \tag{10}\]5The elements \(g_{i}\) may be determined by the following conditions:  
\[\theta^{p-1}=1 \tag{10}\]6  
which follows from Formula (22). Elements \(g_{i}\) for the case under consideration can be selected as follows:  
\[\theta^{p-1}=1 \tag{10}\]7  
as proven by direct inspection. Consequently, the elements of the field-generating sequence (1) can be expressed as follows:  
\[\theta^{p-1}=1 \tag{10}\]8  
where the elements \(g_{i}\) are expressed by Formula (26), and calculations are performed in the field \(GF(31)\). Accordingly, the other elements of sequence (1) are given by the expressions:  
\[\theta^{p-1}=1 \tag{10}\]9  
Formula (28) shows that sequence (1) can be represented as a product of three functions; each of them is given by the degrees of the elements \(g_{i}\). Let us consider the following functions:  
\[R=r_{1}+r_{2}+\cdots+r_{n} \tag{11}\]0  
There can be only two such functions, since \(g_{1}^{2}=1(31)\). Moreover, even for \(m\), the right part of Formula (29) converges to one, i.e., one of these functions is identically equal to one, and the second one is periodic with period 2. Similarly, there are only three subsequent functions:  
\[R=r_{1}+r_{2}+\cdots+r_{n} \tag{11}\]1  
since \(g_{2}^{3}=1(31)\) and one of them is identically equal to one too (this corresponds to the case where \(m\) is a multiple of three). These functions have a period equal to three. Since \(g_{3}^{5}=1(31)\), there are five subsequent functions:  
\[R=r_{1}+r_{2}+\cdots+r_{n} \tag{11}\]2  
One of which is also identically equal to one, and the period of the others is equal to five. We will show that the proposed representation of basic functions allows for the implementation of a rather simple algorithm of the Fourier-Galois spectrum analyzer, and it is oriented toward the use of typical logic elements.