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	Manual: The Kronecker symbol - libzahl - big integer library
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	---
	commit 019da3a9e7f81cd882d0383ac707ce098013b4a9
	parent 60dd5110e21d1aedc047f2033af74330df552e40
	Author: Mattias Andrée <[email protected]>
	Date: Mon, 25 Jul 2016 16:38:43 +0200

	Manual: The Kronecker symbol

	Signed-off-by: Mattias Andrée <[email protected]>

	Diffstat:
	M doc/not-implemented.tex \| 60 +++++++++++++++++++++++++++++…

	1 file changed, 56 insertions(+), 4 deletions(-)
	---
	diff --git a/doc/not-implemented.tex b/doc/not-implemented.tex
	@@ -163,7 +163,8 @@ so a compressed lookup table can be used for small $p$.
	\left ( \frac{a}{n} \right ) =
	\prod_k \left ( \frac{a}{p_k} \right )^{n_k},
	}\)
	-where $n$ = $\displaystyle{\prod_k p_k^{n_k}}$, and $p_k \in \textbf{P}$.
	+where $\displaystyle{n = \prod_k p_k^{n_k} > 0}$,
	+and $p_k \in \textbf{P}$.
	\vspace{1em}

	Like the Legendre symbol, the Jacobi symbol is $n$-period over $a$.
	@@ -197,14 +198,65 @@ Use the following algorithm to calculate the Jacobi symbo…
	\STATE \textbf{start over}
	\end{algorithmic}
	\end{minipage}
	-\vspace{1em}
	-


	\subsection{Kronecker symbol}
	\label{sec:Kronecker symbol}

	-TODO
	+The Kronecker symbol
	+$\displaystyle{\left ( \frac{a}{n} \right )}$
	+is a generalisation of the Jacobi symbol,
	+where $n$ can be any integer. For positive
	+odd $n$, the Kronecker symbol is equal to
	+the Jacobi symbol. For even $n$, the
	+Kronecker symbol is $2n$-periodic over $a$,
	+the Kronecker symbol is zero for all
	+$(a, n)$ with both $a$ and $n$ are even.
	+
	+\vspace{1em}
	+\noindent
	+\( \displaystyle{
	+ \left ( \frac{a}{2^k \cdot n} \right ) =
	+ \left ( \frac{a}{n} \right ) \cdot \left ( \frac{a}{2} \right )^k,
	+}\)
	+where
	+\( \displaystyle{
	+ \left ( \frac{a}{2} \right ) =
	+ \left \lbrace \begin{array}{rl}
	+ 1 & \text{if}~ a \equiv 1, 7 ~(\text{Mod}~ 8) \\
	+ -1 & \text{if}~ a \equiv 3, 5 ~(\text{Mod}~ 8) \\
	+ 0 & \text{otherwise}
	+ \end{array} \right .
	+}\)
	+
	+\vspace{1em}
	+\noindent
	+\( \displaystyle{
	+ \left ( \frac{-a}{n} \right ) =
	+ \left ( \frac{a}{n} \right ) \cdot \left ( \frac{a}{-1} \right ),
	+}\)
	+where
	+\( \displaystyle{
	+ \left ( \frac{a}{-1} \right ) =
	+ \left \lbrace \begin{array}{rl}
	+ 1 & \text{if}~ a \ge 0 \\
	+ -1 & \text{if}~ a < 0
	+ \end{array} \right .
	+}\)
	+\vspace{1em}
	+
	+\noindent
	+However, for $n = 0$, the symbol is defined as
	+
	+\vspace{1em}
	+\noindent
	+\( \displaystyle{
	+ \left ( \frac{a}{0} \right ) =
	+ \left \lbrace \begin{array}{rl}
	+ 1 & \text{if}~ a = \pm 1 \\
	+ 0 & \text{otherwise.}
	+ \end{array} \right .
	+}\)


	\subsection{Power residue symbol}