Часть 1.Простое

представимо формой

, где

целые числа.
Причём

Пусть

один из первообразных корней простого

. Обозначим
![$$ \[
\varsigma _{\left( m \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g^m } } = \left\{ \begin{array}{l}
\varsigma _{\left( 0 \right)} \to m = 0\left( {\bmod 3} \right) \\
\varsigma _{\left( 1 \right)} \to m = 1\left( {\bmod 3} \right) \\
\varsigma _{\left( 2 \right)} \to m = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $ $$ \[
\varsigma _{\left( m \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g^m } } = \left\{ \begin{array}{l}
\varsigma _{\left( 0 \right)} \to m = 0\left( {\bmod 3} \right) \\
\varsigma _{\left( 1 \right)} \to m = 1\left( {\bmod 3} \right) \\
\varsigma _{\left( 2 \right)} \to m = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $](https://dxdy-02.korotkov.co.uk/f/1/c/4/1c4c43bc22efcbb33c252e277ee3975682.png)

![$$ \[
\begin{array}{l}
\eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 \\
\tilde \eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon \\
\end{array}
\] $ $$ \[
\begin{array}{l}
\eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 \\
\tilde \eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon \\
\end{array}
\] $](https://dxdy-04.korotkov.co.uk/f/f/f/2/ff2f3c8a8d1a0e22b04988803c96755d82.png)
![$$ \[
\eta _{\left( n \right)} = \left\{ \begin{array}{l}
\eta _{\left( 0 \right)} \to n = 0\left( {\bmod 3} \right) \\
\eta _{\left( 0 \right)} \varepsilon ^2 \to n = 1\left( {\bmod 3} \right) \\
\eta _{\left( 0 \right)} \varepsilon \to n = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $ $$ \[
\eta _{\left( n \right)} = \left\{ \begin{array}{l}
\eta _{\left( 0 \right)} \to n = 0\left( {\bmod 3} \right) \\
\eta _{\left( 0 \right)} \varepsilon ^2 \to n = 1\left( {\bmod 3} \right) \\
\eta _{\left( 0 \right)} \varepsilon \to n = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $](https://dxdy-01.korotkov.co.uk/f/c/8/c/c8cd125cfd3d205d43c3ea1e8fdba57d82.png)
![$$ \[
\tilde \eta _{\left( n \right)} = \left\{ \begin{array}{l}
\tilde \eta _{\left( 0 \right)} \to n = 0\left( {\bmod 3} \right) \\
\tilde \eta _{\left( 0 \right)} \varepsilon \to n = 1\left( {\bmod 3} \right) \\
\tilde \eta _{\left( 0 \right)} \varepsilon ^2 \to n = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $ $$ \[
\tilde \eta _{\left( n \right)} = \left\{ \begin{array}{l}
\tilde \eta _{\left( 0 \right)} \to n = 0\left( {\bmod 3} \right) \\
\tilde \eta _{\left( 0 \right)} \varepsilon \to n = 1\left( {\bmod 3} \right) \\
\tilde \eta _{\left( 0 \right)} \varepsilon ^2 \to n = 2\left( {\bmod 3} \right) \\
\end{array} \right.
\] $](https://dxdy-02.korotkov.co.uk/f/1/e/4/1e4e854f9e8d5ff97ba5b27b9197e5d282.png)
Для всех

произведение
![$$ \[
\eta _{\left( n \right)} \tilde \eta _{\left( n \right)} = \eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)}
\] $ $$ \[
\eta _{\left( n \right)} \tilde \eta _{\left( n \right)} = \eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)}
\] $](https://dxdy-04.korotkov.co.uk/f/f/e/3/fe31c2d19f01bd915c2b180025a0a27b82.png)
не зависит от

. Иначе, все изоморфизмы произведения в кольце

![$\[
\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $ $\[
\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $](https://dxdy-02.korotkov.co.uk/f/5/7/e/57e1165d613b21e723592ef18ec969e582.png)
равны. Также равны и изоморфизмы произведения в кольце


. Отсюда это произведение равно целому числу. (Теория чисел)
Покажем, что это произведение равно

.
![$$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 } \right)\left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon } \right) =
\] $ $$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 } \right)\left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon } \right) =
\] $](https://dxdy-04.korotkov.co.uk/f/3/3/5/3352b54670257668bf14967722ba374a82.png)
![$$ \[
\left( {\varsigma _{\left( 0 \right)} ^2 + \varsigma _{\left( 1 \right)} ^2 + \varsigma _{\left( 2 \right)} ^2 } \right) - \left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right) =
\] $ $$ \[
\left( {\varsigma _{\left( 0 \right)} ^2 + \varsigma _{\left( 1 \right)} ^2 + \varsigma _{\left( 2 \right)} ^2 } \right) - \left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right) =
\] $](https://dxdy-02.korotkov.co.uk/f/1/4/0/140772a9385e52d35d039353b55bb25e82.png)
![$$ \[
= \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} + \varsigma _{\left( 2 \right)} } \right)^2 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right)
\] $ $$ \[
= \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} + \varsigma _{\left( 2 \right)} } \right)^2 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right)
\] $](https://dxdy-02.korotkov.co.uk/f/d/3/a/d3a9e3a0821d2264ebfee9ace3d9eb6482.png)
Так как
![$$ \[
\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} + \varsigma _{\left( 2 \right)} = - 1
\] $ $$ \[
\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} + \varsigma _{\left( 2 \right)} = - 1
\] $](https://dxdy-03.korotkov.co.uk/f/6/b/e/6bec92cad62058bd94c0ffc67a41bcd482.png)
то
![$$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = 1 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right)
\] $ $$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = 1 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 0 \right)} } \right)
\] $](https://dxdy-04.korotkov.co.uk/f/b/9/2/b92af39418e04ab8e87bb0d3bcf028ff82.png)
Выражение в скобках равно целому числу, так как все изоморфизмы равны.
Из теории.
Пусть известно, что сумма
![$ \[\sum\limits_{k = 1}^t {e^{\frac{{2\pi i}}{P}a_k } } \] $ $ \[\sum\limits_{k = 1}^t {e^{\frac{{2\pi i}}{P}a_k } } \] $](https://dxdy-04.korotkov.co.uk/f/f/3/6/f36f0a182c4da0fc6021d4a0611ba68e82.png)
равна целому числу и все

целые числа. Тогда сумму можно представить
![$$ \[
\sum\limits_{k = 1}^t {e^{\frac{{2\pi i}}{P}a_k } } = n + m\sum\limits_{k = 1}^P {e^{\frac{{2\pi i}}{P}k} } = n - m
\] $ $$ \[
\sum\limits_{k = 1}^t {e^{\frac{{2\pi i}}{P}a_k } } = n + m\sum\limits_{k = 1}^P {e^{\frac{{2\pi i}}{P}k} } = n - m
\] $](https://dxdy-04.korotkov.co.uk/f/f/b/3/fb35d123c3671cd3795e9286ac7af30282.png)
где

число показателей

для которых
![$\[a_k \equiv 0\left( {\bmod P} \right)\]$ $\[a_k \equiv 0\left( {\bmod P} \right)\]$](https://dxdy-02.korotkov.co.uk/f/9/b/3/9b34e8bfb23209eeef0f31119ef65c7282.png)
Итак, в скобках выражение равно целому числу, число слагаемых равно
![$ \[
3\left( {\frac{{P - 1}}{3}} \right)^2 = \frac{{\left( {P - 1} \right)}}{3}^2 \]$ $ \[
3\left( {\frac{{P - 1}}{3}} \right)^2 = \frac{{\left( {P - 1} \right)}}{3}^2 \]$](https://dxdy-04.korotkov.co.uk/f/f/d/1/fd1a579480b2789a8281b4cdc108d63f82.png)
Найдём число членов с показателем, делящимся на

.
![$$ \[
\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} = \sum\limits_{k,t = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} } e^{\frac{{2\pi i}}{P}g^{3t} g} } = \sum\limits_{k,t = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}\left( {g^{3k} + g^{3t + 1} } \right)} }
\] $ $$ \[
\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} = \sum\limits_{k,t = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} } e^{\frac{{2\pi i}}{P}g^{3t} g} } = \sum\limits_{k,t = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}\left( {g^{3k} + g^{3t + 1} } \right)} }
\] $](https://dxdy-02.korotkov.co.uk/f/d/f/7/df7a5881a65c6b79e0003a06ac78b36982.png)
![$$ \[
g^{3k} + g^{3t + 1} \equiv 0\left( {\bmod P} \right) \to 1 + g^{3\left( {t - k} \right) + 1} \equiv 0\left( {\bmod P} \right)
\] $ $$ \[
g^{3k} + g^{3t + 1} \equiv 0\left( {\bmod P} \right) \to 1 + g^{3\left( {t - k} \right) + 1} \equiv 0\left( {\bmod P} \right)
\] $](https://dxdy-04.korotkov.co.uk/f/7/0/8/70820e2228a6bca10be4833b7b1240ce82.png)
Это последнее равенство возможно только если
![$$ \[ 3\left( {t - k} \right) + 1 = \left( {2s + 1} \right)\frac{{P - 1}}{2}\] $ $$ \[ 3\left( {t - k} \right) + 1 = \left( {2s + 1} \right)\frac{{P - 1}}{2}\] $](https://dxdy-01.korotkov.co.uk/f/0/0/4/004dec875cf711987247315f0ea195b682.png)
что невозможно. Следовательно, все слагаемые отличны от единицы. Аналогично и для двух других слагаемых в скобке.
Тогда
![$$ \[
\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 1 \right)} = \sum\limits_{i = 1}^{\frac{{\left( {P - 1} \right)^2 }}{3}} {e^{\frac{{2\pi i}}{P}m_i } } = \frac{{P - 1}}{3}\sum\limits_1^{P - 1} {e^{\frac{{2\pi i}}{P}m} } = - \frac{{P - 1}}{3}
\] $ $$ \[
\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 1 \right)} = \sum\limits_{i = 1}^{\frac{{\left( {P - 1} \right)^2 }}{3}} {e^{\frac{{2\pi i}}{P}m_i } } = \frac{{P - 1}}{3}\sum\limits_1^{P - 1} {e^{\frac{{2\pi i}}{P}m} } = - \frac{{P - 1}}{3}
\] $](https://dxdy-03.korotkov.co.uk/f/2/4/1/241c004c4f43594600dea0d12e8e24de82.png)
![$$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = 1 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 1 \right)} } \right) = P
\] $ $$ \[
\eta _{\left( 0 \right)} \tilde \eta _{\left( 0 \right)} = 1 - 3\left( {\varsigma _{\left( 0 \right)} \varsigma _{\left( 1 \right)} + \varsigma _{\left( 1 \right)} \varsigma _{\left( 2 \right)} + \varsigma _{\left( 2 \right)} \varsigma _{\left( 1 \right)} } \right) = P
\] $](https://dxdy-02.korotkov.co.uk/f/d/2/c/d2cd7eb17881e8474cb965e1b86fc4f782.png)
===
-- Сб май 03, 2014 20:21:52 --Часть 2.Из теории.
Если для целого числа в кольце

![$\[\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $ $\[\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $](https://dxdy-01.korotkov.co.uk/f/8/c/4/8c4800f21966500ad44d221b0bbac9ca82.png)
![$ \[ d \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right) \] $ $ \[ d \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right) \] $](https://dxdy-02.korotkov.co.uk/f/d/f/d/dfd5fb1d989475af2d9eaf386114a8ac82.png)
, то

Если для многочлена с целыми коэффициентами

, то

Рассмотрим выражение
![$$ \[
\eta _{\left( 0 \right)} ^3 = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 } \right)^3
\] $ $$ \[
\eta _{\left( 0 \right)} ^3 = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 } \right)^3
\] $](https://dxdy-03.korotkov.co.uk/f/e/e/6/ee6c2854045052615b82a56d7c704e5382.png)
Все его изоморфизмы в кольце

![$\[\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $ $\[\left( {e^{\frac{{2\pi i}}{P}} } \right) \] $](https://dxdy-01.korotkov.co.uk/f/8/c/4/8c4800f21966500ad44d221b0bbac9ca82.png)
равны.
Следовательно

-целые,

- (Из теории)
![$$ \[
\tilde \eta _{\left( 0 \right)} ^3 = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon } \right)^3 = c\left( {a + b\varepsilon ^2 } \right)
\] $ $$ \[
\tilde \eta _{\left( 0 \right)} ^3 = \left( {\varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon } \right)^3 = c\left( {a + b\varepsilon ^2 } \right)
\] $](https://dxdy-04.korotkov.co.uk/f/f/7/b/f7b6fe63812c2b6011e157667b3ba12382.png)
![$$ \[
\eta _{\left( 0 \right)} ^3 \tilde \eta _{\left( 0 \right)} ^3 = P^3 = c^2 \left( {a + b\varepsilon } \right)\left( {a + b\varepsilon ^2 } \right) = c^2 \left( {a^2 - ab + b^2 } \right) \to c = 1,P
\] $ $$ \[
\eta _{\left( 0 \right)} ^3 \tilde \eta _{\left( 0 \right)} ^3 = P^3 = c^2 \left( {a + b\varepsilon } \right)\left( {a + b\varepsilon ^2 } \right) = c^2 \left( {a^2 - ab + b^2 } \right) \to c = 1,P
\] $](https://dxdy-01.korotkov.co.uk/f/8/3/4/834e3232fbc632d0dbeb8c1b86b7bf0f82.png)
Пусть

Рассматривая

как многочлен от

имеем
![$$ \[
\eta _{\left( 0 \right)} \left( 1 \right) = \frac{{P - 1}}{3} + \frac{{P - 1}}{3}\varepsilon + \frac{{P - 1}}{3}\varepsilon ^2 = 0 \equiv 0\left( {\bmod P} \right)
\] $ $$ \[
\eta _{\left( 0 \right)} \left( 1 \right) = \frac{{P - 1}}{3} + \frac{{P - 1}}{3}\varepsilon + \frac{{P - 1}}{3}\varepsilon ^2 = 0 \equiv 0\left( {\bmod P} \right)
\] $](https://dxdy-02.korotkov.co.uk/f/9/d/b/9dbb3da75d91786023b69423c312695e82.png)
Следовательно
![$$ \[
\eta _{\left( 0 \right)} \equiv \tilde \eta _{\left( 0 \right)} \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $ $$ \[
\eta _{\left( 0 \right)} \equiv \tilde \eta _{\left( 0 \right)} \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $](https://dxdy-02.korotkov.co.uk/f/9/1/1/911d02789820966fd03ed18abafdb7a482.png)
![$$ \[
\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 = b\left( {\varepsilon - \varepsilon ^2 } \right) \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $ $$ \[
\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 = b\left( {\varepsilon - \varepsilon ^2 } \right) \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $](https://dxdy-01.korotkov.co.uk/f/8/e/3/8e3e3a5d2fa08538380e3f2d1ccfeb5982.png)
![$$ \[
\left( {\varepsilon ^2 - \varepsilon } \right)\left( {\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 } \right) = 3b \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $ $$ \[
\left( {\varepsilon ^2 - \varepsilon } \right)\left( {\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 } \right) = 3b \equiv 0\left( {\bmod \left( {1 - e^{\frac{{2\pi i}}{P}} } \right)} \right)
\] $](https://dxdy-01.korotkov.co.uk/f/c/2/4/c242b46e6e4cf15afcc38c796d7872e882.png)
Отсюда

А значит и

должно делится на

, что невозможно по взаимной простоте

.
Итак,


=====
-- Сб май 03, 2014 20:23:29 --Часть 3.Окончательно получим:

- простое,

- его первообразный корень.

![$$ \[
\varsigma _{\left( 0 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} } } ,\varsigma _{\left( 1 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g} } ,\varsigma _{\left( 2 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g^2 } }
\] $ $$ \[
\varsigma _{\left( 0 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} } } ,\varsigma _{\left( 1 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g} } ,\varsigma _{\left( 2 \right)} = \sum\limits_{k = 1}^{\frac{{P - 1}}{3}} {e^{\frac{{2\pi i}}{P}g^{3k} g^2 } }
\] $](https://dxdy-02.korotkov.co.uk/f/5/2/9/5292f32971690e7f110ddeb18e0ab06682.png)
![$$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 \\
\tilde \eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon \\
\end{array} \right.
\] $ $$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon + \varsigma _{\left( 2 \right)} \varepsilon ^2 \\
\tilde \eta _{\left( 0 \right)} = \varsigma _{\left( 0 \right)} + \varsigma _{\left( 1 \right)} \varepsilon ^2 + \varsigma _{\left( 2 \right)} \varepsilon \\
\end{array} \right.
\] $](https://dxdy-02.korotkov.co.uk/f/1/8/6/1867d46a308d125f1b1d143ba2bb3fe382.png)
![$$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} ^3 = P\left( {a + b\varepsilon } \right) \\
\tilde \eta _{\left( 0 \right)} ^3 = P\left( {a + b\varepsilon ^2 } \right) \\
\end{array} \right.
\]$ $$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} ^3 = P\left( {a + b\varepsilon } \right) \\
\tilde \eta _{\left( 0 \right)} ^3 = P\left( {a + b\varepsilon ^2 } \right) \\
\end{array} \right.
\]$](https://dxdy-01.korotkov.co.uk/f/4/0/a/40ac6a28366942821412452950d433e082.png)
![$$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 = Pb\left( {\varepsilon - \varepsilon ^2 } \right) \\
\eta _{\left( 0 \right)} ^3 \varepsilon - \tilde \eta _{\left( 0 \right)} ^3 = Pa\left( {\varepsilon - 1} \right) \\
\end{array} \right.
\]$ $$\[
\left\{ \begin{array}{l}
\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 = Pb\left( {\varepsilon - \varepsilon ^2 } \right) \\
\eta _{\left( 0 \right)} ^3 \varepsilon - \tilde \eta _{\left( 0 \right)} ^3 = Pa\left( {\varepsilon - 1} \right) \\
\end{array} \right.
\]$](https://dxdy-02.korotkov.co.uk/f/d/9/3/d9355118ee4560dc4e0f0103d60c848682.png)
![$$\[
a = \frac{{\eta _{\left( 0 \right)} ^3 \varepsilon - \tilde \eta _{\left( 0 \right)} ^3 }}{{P\left( {\varepsilon - 1} \right)}},b = \frac{{\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 }}{{P\left( {\varepsilon - \varepsilon ^2 } \right)}}
\]$ $$\[
a = \frac{{\eta _{\left( 0 \right)} ^3 \varepsilon - \tilde \eta _{\left( 0 \right)} ^3 }}{{P\left( {\varepsilon - 1} \right)}},b = \frac{{\eta _{\left( 0 \right)} ^3 - \tilde \eta _{\left( 0 \right)} ^3 }}{{P\left( {\varepsilon - \varepsilon ^2 } \right)}}
\]$](https://dxdy-02.korotkov.co.uk/f/d/3/b/d3b4806a7c150a855e885c680df249a282.png)