ID

Symbol

Reducement

1

$\stackrel{3/2/o=1}{\overbrace{\left\{\mathbb{S}\right\}}}$

2

$\stackrel{2/1}{\overbrace{\left\{\mathbb{S}\right\}}}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}7& \begin{array}{cc}19& \begin{array}{cc}37& \begin{array}{cc}61& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \cdots \end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{cc}1& \begin{array}{cc}6& \begin{array}{cc}12& \begin{array}{cc}18& \begin{array}{cc}24& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$

3

$\stackrel{1/0}{\overbrace{\left\{\mathbb{S}\right\}}}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}6& \begin{array}{cc}12& \begin{array}{cc}18& \begin{array}{cc}24& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \cdots \end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{cc}1& \begin{array}{cc}5& \begin{array}{cc}6& \begin{array}{cc}6& \begin{array}{cc}6& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$

4

$\stackrel{0/\text{kernel}}{\overbrace{\left\{\mathbb{S}\right\}}}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}5& \begin{array}{cc}6& \begin{array}{cc}6& \begin{array}{cc}6& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \cdots \end{array}\end{array}\end{array}\right\}\\ =\langle \begin{array}{cc}1& \begin{array}{cc}4& \begin{array}{cc}1& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\rangle \end{array}$

5

$\stackrel{0/-1}{\overbrace{\left\{\mathbb{S}\right\}}}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}4& \begin{array}{cc}1& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& -1\end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}-3& \begin{array}{cc}-1& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}=\left\{\begin{array}{cc}1& \begin{array}{cc}4& \begin{array}{cc}1& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\right\}\end{array}$

6

$\stackrel{-1/-2}{\overbrace{\left\{\mathbb{S}\right\}}}$

$\begin{array}{l}\left\{\begin{array}{cc}1& \begin{array}{cc}3& \begin{array}{cc}-3& \begin{array}{cc}-1& \begin{array}{cc}0& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& \begin{array}{cc}\begin{array}{c}+1\\ -1\end{array}& -1\end{array}\end{array}\end{array}\end{array}\right\}\\ =\left\{\begin{array}{cc}1& \begin{array}{cc}2& \begin{array}{cc}-6& \begin{array}{cc}2& \begin{array}{cc}1& 0\end{array}\end{array}\end{array}\end{array}\end{array}\right\}\end{array}$

7

$\stackrel{3/\ker }{\overbrace{\left\{\mathbb{S}\right\}}}$

$\stackrel{\text{Threestepsinoneandhencethescopeforgeneralisation}}{\overbrace{\left\{\begin{array}{cc}1& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \cdots \end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}1\\ -4\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ 6\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ -4\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ \begin{array}{c}-4\\ 1\end{array}\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ \begin{array}{c}-4\\ 1\end{array}\end{array}\end{array}\end{array}& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}}}=\langle \begin{array}{cc}1& \begin{array}{cc}4& 1\end{array}\end{array}\rangle$

8

$\stackrel{\text{Fourstepsinoneandhencethescopeforgeneralisation}}{\overbrace{\left\{\begin{array}{cc}0& \begin{array}{cc}8& \begin{array}{cc}27& \begin{array}{cc}64& \cdots \end{array}\end{array}\end{array}\end{array}\right\}\oplus \left\{0\begin{array}{cc}1& \begin{array}{cc}\begin{array}{c}1\\ -4\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ 6\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ -4\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ \begin{array}{c}-4\\ 1\end{array}\end{array}\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}1\\ \begin{array}{c}-4\\ \begin{array}{c}6\\ \begin{array}{c}-4\\ 1\end{array}\end{array}\end{array}\end{array}& \cdots \end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right\}}}$

$=\left\{\begin{array}{ccc}0& 8& \begin{array}{ccc}-5& 4& -1\end{array}\end{array}\right\}$