 | ■No23119に返信(ミニーさんの記事)
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解決済み! でしょうが
Ｑの拡大体Ｑ（Sqrt[3]+Sqrt[5]）達の視座から ；
In[2]:=
f1 = x1^2 - 3;
f2 = x2^2 - 5;
f = x - (x1 + x2);
GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}]
Solve[% == {0, 0, 0}];
GB[[1]]
Out[5]=
{4 - 16*x^2 + x^4, 18*x - x^3 - 4*x2,
14*x - x^3 + 4*x1}
Out[7]=
4 - 16*x^2 + x^4
In[8]:=
f = GB[[1]]; g = x;
{f, g}
{d, {X0, Y0}} = PolynomialExtendedGCD[f, g]
{{f*X0}, {g*Y0}}
Out[9]=
{4 - 16*x^2 + x^4, x}
Out[10]=
{1, {1/4, 4*x - x^3/4}}
Out[11]=
{{1/4*(4 - 16*x^2 + x^4)},
{x*(4*x - x^3/4)}}
In[12]:=
Y0 /. x -> Sqrt[3] + Sqrt[5]
Expand[%]
A1 = Expand[%*2]
N[%]
Out[12]=
4*(Sqrt[3] + Sqrt[5]) -
1/4*(Sqrt[3] + Sqrt[5])^3
Out[13]=
-(Sqrt[3]/2) + Sqrt[5]/2
Out[14]=
-Sqrt[3] + Sqrt[5]
Out[15]=
0.5040171699309126
In[16]:=
f1 = x1^2 - 5;
f2 = x2^2 - 7;
f = x - (x1 + x2);
GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}]
Solve[% == {0, 0, 0}];
GB[[1]]
Out[19]=
{4 - 24*x^2 + x^4, 26*x - x^3 - 4*x2,
22*x - x^3 + 4*x1}
Out[21]=
4 - 24*x^2 + x^4
In[22]:=
f = GB[[1]]; g = x;
{f, g}
{d, {X0, Y0}} = PolynomialExtendedGCD[f, g]
{{f*X0}, {g*Y0}}
Out[23]=
{4 - 24*x^2 + x^4, x}
Out[24]=
{1, {1/4, 6*x - x^3/4}}
Out[25]=
{{1/4*(4 - 24*x^2 + x^4)},
{x*(6*x - x^3/4)}}
In[26]:=
Y0 /. x -> Sqrt[5] + Sqrt[7]
Expand[%]
A2 = Expand[%*2]
N[%]
Out[26]=
6*(Sqrt[5] + Sqrt[7]) -
1/4*(Sqrt[5] + Sqrt[7])^3
Out[27]=
-(Sqrt[5]/2) + Sqrt[7]/2
Out[28]=
-Sqrt[5] + Sqrt[7]
Out[29]=
0.4096833335648009
In[30]:=
f1 = x1^2 - 7;
f2 = x2^2 - 9;
f = x - (x1 + x2);
GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}]
Solve[% == {0, 0, 0}];
GB[[1]]
Out[33]=
{4 - 32*x^2 + x^4, 34*x - x^3 - 4*x2,
30*x - x^3 + 4*x1}
Out[35]=
4 - 32*x^2 + x^4
In[36]:=
f = GB[[1]]; g = x;
{f, g}
{d, {X0, Y0}} = PolynomialExtendedGCD[f, g]
{{f*X0}, {g*Y0}}
Out[37]=
{4 - 32*x^2 + x^4, x}
Out[38]=
{1, {1/4, 8*x - x^3/4}}
Out[39]=
{{1/4*(4 - 32*x^2 + x^4)},
{x*(8*x - x^3/4)}}
In[40]:=
Y0 /. x -> Sqrt[7] + Sqrt[9]
Expand[%]
A3 = Expand[%*2]
N[%]
Out[40]=
8*(3 + Sqrt[7]) - 1/4*(3 + Sqrt[7])^3
Out[41]=
3/2 - Sqrt[7]/2
Out[42]=
3 - Sqrt[7]
Out[43]=
0.3542486889354093
In[44]:=
f1 = x1^2 - 9;
f2 = x2^2 - 11;
f = x - (x1 + x2);
GB = GroebnerBasis[{f1, f2, f}, {x1, x2, x}]
Solve[% == {0, 0, 0}];
GB[[1]]
Out[47]=
{4 - 40*x^2 + x^4, 42*x - x^3 - 4*x2,
38*x - x^3 + 4*x1}
Out[49]=
4 - 40*x^2 + x^4
In[50]:=
f = GB[[1]]; g = x;
{f, g}
{d, {X0, Y0}} = PolynomialExtendedGCD[f, g]
{{f*X0}, {g*Y0}}
Out[51]=
{4 - 40*x^2 + x^4, x}
Out[52]=
{1, {1/4, 10*x - x^3/4}}
Out[53]=
{{1/4*(4 - 40*x^2 + x^4)},
{x*(10*x - x^3/4)}}
In[54]:=
Y0 /. x -> Sqrt[9] + Sqrt[11]
Expand[%]
A4 = Expand[%*2]
N[%]
Out[54]=
10*(3 + Sqrt[11]) - 1/4*(3 + Sqrt[11])^3
Out[55]=
-(3/2) + Sqrt[11]/2
Out[56]=
-3 + Sqrt[11]
Out[57]=
0.3166247903553998
In[58]:=
{A1, A2, A3, A4}
Out[58]=
{-Sqrt[3] + Sqrt[5], -Sqrt[5] + Sqrt[7],
3 - Sqrt[7], -3 + Sqrt[11]}
In[59]:=
A1 + A2 + A3 + A4
N[%]
Out[59]=
-Sqrt[3] + Sqrt[11]<--コタエ
Out[60]=
1.5845739827865226
時には ▼分母に無理数がない▼ 子のように
誰にもI(deal)を話せない... なんて
商環 k[X]/I I(deal)は地球を救う とか 巷で
手ごろのが在るので；
http://blog.livedoor.jp/seven_triton/archives/2006-05.html
様 の √素数の問題 の 頁 に; 問４をどうぞ。
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