Eisenstein Integer

Author

Eric W. Weisstein
February 24, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/EisensteinInteger.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EisensteinInteger.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Multiplication

ComplexExpand[Through[{Re,Im}[Expand[(a+b ω)(c+d ω)]]]]//Factor

Expand[(a+b ω)(c+d ω)]

FullSimplify[(a+b ω)(a+b (ω^2))/.ω->(1/2) (-1+I (Sqrt[3]))]

Normal Form

OmegaNormalForm[expr_]:=Module[{e=expr/.ω->(1/2) (-1+I (Sqrt[3])),c},
c=Coefficient[Expand[e],Sqrt[3]]2/I;
c ω+RootReduce[e-c (1/2) (-1+I (Sqrt[3]))]
]

Units

<<NumberTheory`AlgebraicNumberFields`

units=RootsOfUnity[Sqrt[-3]]

RootReduce[units]

units^6

"Positive" Numbers

points=
Flatten[Table[{a,b},{a,-10,10},{b,-10,10}],1];

<<Graphics`Colors`

With[{w=(1/2) (-1+I (Sqrt[3]))},
Show[Graphics[{
{Hue[.5],Polygon[{0,.2}+#&/@(5{{0,0},Through[{Re,Im}[1-w]],Through[{Re,Im}[2+w]]})]
},
Line[{{-5,0},{5,0}}],
Line[{{0,-5},{0,5}}],
Text[TraditionalForm[{1,ω}.#],{Re[#.{1,w}],Im[#.{1,w}]}+{0,-.2},TextStyle->{FontSize->8},Background->If[Re[{1,w}.#]==0,White,None]
]&/@points,
PointSize[.02],
Point/@({Re[#],Im[#]}&/@({1,w}.#&/@points))
}
],AspectRatio->Automatic,PlotRange->{{-4,4},{-4,4}}]
]

Dörrie's Form

j=(1+I Sqrt[3])/2;
o=(1-I Sqrt[3])/2;

Simplify[{j+o,j o,j^2+o,o^2+j,j^3,o^3}]