Semi-system

A half-system modulo is a subset of , the set of the various remainder classes modulo , in which exactly either or lies for each . Given a half- system, the complementary half-system is called . ${\displaystyle n}$ ${\displaystyle X:=(\mathbb {Z} /n\mathbb {Z} )\setminus \{{\bar {0}}\}}$ ${\displaystyle {\bar {0}}}$ ${\displaystyle n}$ ${\displaystyle x\in X}$ ${\displaystyle x}$ ${\displaystyle -x}$ ${\displaystyle H}$ ${\displaystyle X\setminus H}$ ${\displaystyle H'}$

Half systems are used by Leopold Kronecker's access to the Jacobi symbol .

example

In the cyclic residue class group modulo an odd prime number , for example, the following quantity is a half System: ${\displaystyle p}$ ${\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{\times }=(\mathbb {Z} /p\mathbb {Z} )\setminus \{{\bar {0}}\}}$

${\displaystyle H:=\left\{g^{k}\mid 1\leq k\leq {\frac {p-1}{2}}\right\}}$

${\displaystyle g}$here denotes a primitive root of the cyclic group. Proof: contains just half of the elements of , which itself contains elements. There is is for the negative . ${\displaystyle H}$ ${\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{\times }}$ ${\displaystyle p-1}$ ${\displaystyle {\overline {-1}}=g^{\frac {p-1}{2}}}$ ${\displaystyle g^{k}=x\in H}$ ${\displaystyle -x=-g^{k}=g^{\frac {p-1}{2}}g^{k}=g^{{\frac {p-1}{2}}+k}\notin H}$

literature

• Armin Leutbecher, number theory . Springer-Verlag, 1996. ISBN 3-540-58791-8 .