Zum Inhalt springen

# Kreisteilungspolynom/Zerlegung mod p/Tabelle

Die folgende Tabelle zeigt die Primfaktorzerlegung der (über ${\displaystyle {}\mathbb {Z} }$ und ${\displaystyle {}\mathbb {Q} }$ irreduziblen) Kreisteilungspolynome ${\displaystyle {}\Phi _{n}}$ in ${\displaystyle {}\mathbb {Z} /(p)[X]}$ für die ersten Primzahlen ${\displaystyle {}p=2,3,5,7,11,13}$.

n ${\displaystyle {}\Phi _{n}}$ 2 3 5 7 11 13
1 ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$ ${\displaystyle {}X-1}$
2 ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$ ${\displaystyle {}X+1}$
3 ${\displaystyle {}X^{2}+X+1}$ ${\displaystyle {}X^{2}+X+1}$ ${\displaystyle {}(X+2)^{2}}$ ${\displaystyle {}X^{2}+X+1}$ ${\displaystyle {}(X+5)(X+3)}$ ${\displaystyle {}X^{2}+X+1}$ ${\displaystyle {}(X+10)(X+4)}$
4 ${\displaystyle {}X^{2}+1}$ ${\displaystyle {}(X+1)^{2}}$ ${\displaystyle {}X^{2}+1}$ ${\displaystyle {}(X+2)(X+3)}$ ${\displaystyle {}X^{2}+1}$ ${\displaystyle {}X^{2}+1}$ ${\displaystyle {}(X+5)(X+8)}$
5 ${\displaystyle {}X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}(X+4)^{4}}$ ${\displaystyle {}X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}(X+2)(X+6)(X+7)(X+8)}$ ${\displaystyle {}X^{4}+X^{3}+X^{2}+X+1}$
6 ${\displaystyle {}X^{2}-X+1}$ ${\displaystyle {}X^{2}-X+1}$ ${\displaystyle {}(X+1)^{2}}$ ${\displaystyle {}X^{2}-X+1}$ ${\displaystyle {}(X+2)(X+4)}$ ${\displaystyle {}X^{2}-X+1}$ ${\displaystyle {}(X+3)(X+9)}$
7 ${\displaystyle {}X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}(X^{3}+X+1)(X^{3}+X^{2}+1)}$ ${\displaystyle {}X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}X^{6}+X^{5}+X^{4}+X^{3}+X^{2}+X+1}$ ${\displaystyle {}(X+6)^{6}}$ ${\displaystyle {}(X^{3}+5X^{2}+4X+10)(X^{3}+7X^{2}+6X+10)}$ ${\displaystyle {}(X^{2}+3X+1)(X^{2}+5X+1)(X^{2}+6X+1)}$
8 ${\displaystyle {}X^{4}+1}$ ${\displaystyle {}(X+1)^{4}}$ ${\displaystyle {}(X^{2}+2X+2)(X^{2}+X+2)}$ ${\displaystyle {}(X^{2}+3)(X^{2}+2)}$ ${\displaystyle {}(X^{2}+4X+1)(X^{2}+3X+1)}$ ${\displaystyle {}(X^{2}+8X+10)(X^{2}+3X+10)}$ ${\displaystyle {}(X^{2}+8)(X^{2}+5)}$
9 ${\displaystyle {}X^{6}+X^{3}+1}$ ${\displaystyle {}X^{6}+X^{3}+1}$ ${\displaystyle {}(X+2)^{6}}$ ${\displaystyle {}X^{6}+X^{3}+1}$ ${\displaystyle {}(X^{3}+5)(X^{3}+3)}$ ${\displaystyle {}X^{6}+X^{3}+1}$ ${\displaystyle {}(X^{3}+10)(X^{3}+4)}$
10 ${\displaystyle {}X^{4}-X^{3}+X^{2}-X+1}$ ${\displaystyle {}X^{4}-X^{3}+X^{2}-X+1}$ ${\displaystyle {}X^{4}-X^{3}+X^{2}-X+1}$ ${\displaystyle {}(X+1)^{4}}$ ${\displaystyle {}X^{4}-X^{3}+X^{2}-X+1}$ ${\displaystyle {}(X+9)(X+5)(X+4)(X+3)}$ ${\displaystyle {}X^{4}-X^{3}+X^{2}-X+1}$
12 ${\displaystyle {}X^{4}-X^{2}+1}$ ${\displaystyle {}(X^{2}+X+1)^{2}}$ ${\displaystyle {}(X^{2}+1)^{2}}$ ${\displaystyle {}(X^{2}+2X+4)(X^{2}+3X+4)}$ ${\displaystyle {}(X^{2}+4)(X^{2}+2)}$ ${\displaystyle {}(X^{2}+6X+1)(X^{2}+5X+1)}$ ${\displaystyle {}(X+11)(X+7)(X+6)(X+2)}$
15 ${\displaystyle {}X^{8}-X^{7}+X^{5}-X^{4}+X^{3}-X+1}$ ${\displaystyle {}(X^{4}+X+1)(X^{4}+X^{3}+1)}$ ${\displaystyle {}(X^{4}+X^{3}+X^{2}+X+1)^{2}}$ ${\displaystyle {}(X^{2}+X+1)^{4}}$ ${\displaystyle {}(X^{4}+4X^{3}+2X^{2}+X+4)(X^{4}+2X^{3}+4X^{2}+X+2)}$ ${\displaystyle {}(X^{2}+3X+9)(X^{2}+9X+4)(X^{2}+4X+5)(X^{2}+5X+3)}$ ${\displaystyle {}(X^{4}+3X^{3}+9X^{2}+X+3)(X^{4}+9X^{3}+3X^{2}+X+9)}$