Übung Algorithmische Algebra, 24.10.2002

•Aufgabe 2

•a) ggT(a, b mod a)

a = 15 ; b = 2 ; Table[{{a, b - i a}, GCD[a, b - i a]}, {i, -10, 10}]

{{{15, 152}, 1}, {{15, 137}, 1}, {{15, 122}, 1}, {{15, 107}, 1}, {{15, 92}, 1}, {{15, 77}, 1}, {{15, 62}, 1}, {{15, 47}, 1}, {{15, 32}, 1}, {{15, 17}, 1}, {{15, 2}, 1}, {{15, -13}, 1}, {{15, -28}, 1}, {{15, -43}, 1}, {{15, -58}, 1}, {{15, -73}, 1}, {{15, -88}, 1}, {{15, -103}, 1}, {{15, -118}, 1}, {{15, -133}, 1}, {{15, -148}, 1}}

•b) ggT(a, a mod b)

PlotGCD[a_Integer, b_Integer, k_Integer: 20, join_: False] := ListPlot[Table[GCD[a, a - i b], {i, -k, k}], PlotJoined -> join, AxesOrigin -> {k + 1, 0}, PlotRange -> All]

a = 17 ; b = 2 ; GCD[a, b] PlotGCD[a, b, 20, True]

1

[Graphics:HTMLFiles/ue1_6.gif]

-Graphics -

•Aufgabe 3

ζ[n_Integer] := Exp[π i/n]

ζ[3]^3

-1

Table[ζ[3]^i, {i, 0, 6}]

{1, e^(i π)/3, e^(2 i π)/3, -1, e^(-(2 i π)/3), e^(-(i π)/3), 1}

FList[n_Integer] := Table[(X - ζ[n] ^(2 i + 1) Y), {i, 0, n - 1}] F[n_Integer] := Simplify[Product[(X - ζ[n] ^(2 i + 1) Y), {i, 0, n - 1}]]

ζ[8] FList[8] // Simplify

e^(i π)/8

{X - (-1)^(1/8) Y, X - (-1)^(3/8) Y, X - (-1)^(5/8) Y, X - (-1)^(7/8) Y, X + (-1)^(1/8) Y, X + (-1)^(3/8) Y, X + (-1)^(5/8) Y, X + (-1)^(7/8) Y}

F[3]

X^3 + Y^3

FList[16]

{X - e^(i π)/16 Y, X - e^(3 i π)/16 Y, X - e^(5 i π)/16 Y, X - e^(7 i π)/16 Y, X - e^(9 i π)/16 Y, X - e^(11 i π)/16 Y, X - e^(13 i π)/16 Y, X - e^(15 i π)/16 Y, X - e^(-(15 i π)/16) Y, X - e^(-(13 i π)/16) Y, X - e^(-(11 i π)/16) Y, X - e^(-(9 i π)/16) Y, X - e^(-(7 i π)/16) Y, X - e^(-(5 i π)/16) Y, X - e^(-(3 i π)/16) Y, X - e^(-(i π)/16) Y}

F[16]

X^16 + Y^16

•Ideale

•Hauptideale

ZIdeal[a_Integer, r_: 10] := Table[a i , {i, -Quotient[r, 2], Quotient[r, 2]}]

ZIdeal[4]

{-20, -16, -12, -8, -4, 0, 4, 8, 12, 16, 20}

•Durchschnitt von Idealen

ZIdeal[4, 50] ∩ ZIdeal[3, 50]

{-72, -60, -48, -36, -24, -12, 0, 12, 24, 36, 48, 60, 72}

ZIdeal[12]

{-60, -48, -36, -24, -12, 0, 12, 24, 36, 48, 60}

ZIdeal[15, 50] ∩ ZIdeal[21, 50]

{-315, -210, -105, 0, 105, 210, 315}

ZIdeal[LCM[15, 21]]

{-525, -420, -315, -210, -105, 0, 105, 210, 315, 420, 525}

•Vereinigung

ZIdeal[4] ∪ ZIdeal[3]

{-20, -16, -15, -12, -9, -8, -6, -4, -3, 0, 3, 4, 6, 8, 9, 12, 15, 16, 20}

•Multiplikation

Outer[Times, ZIdeal[5], ZIdeal[15]] // Flatten // Union

{-1875, -1500, -1200, -1125, -900, -750, -675, -600, -450, -375, -300, -225, -150, -75, 0, 75, 150, 225, 300, 375, 450, 600, 675, 750, 900, 1125, 1200, 1500, 1875}

•Durch mehrere Elemente erzeugte Ideale

ZIdeal[{a_Integer, b_Integer},    r_: 5] := Block[{lb = Quotient[r, 2], i, j}, Union [Flatten[Table[a i + b j , {i, -lb, lb}, {j, -lb, lb}]]] ]

ZIdeal[{3, 4}]

{-14, -11, -10, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 14}

ZIdeal[{2, 4}]

{-12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12}

ZIdeal[{6, 8}]

{-28, -22, -20, -16, -14, -12, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 28}

ZIdeal[{a_Integer, b_Integer, c_Integer},    r_: 5] := Block[{lb = Quotient[r, 2], i, j, k}, Union [Flatten[Table[a i + b j + c k, {i, -lb, lb}, {j, -lb, lb}, {k, -lb, lb}]]] ]

ZIdeal[{3, 4, 5}]

{-24, -21, -20, -19, -18, -17, -16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24}

ZIdeal[{30, 42, 54}]

{-252, -222, -210, -198, -192, -180, -168, -162, -156, -150, -144, -138, -132, -126, -120, -114, -108, -102, -96, -90, -84, -78, -72, -66, -60, -54, -48, -42, -36, -30, -24, -18, -12, -6, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 180, 192, 198, 210, 222, 252}

Converted by Mathematica  (November 19, 2002)