ToNumberField isn't perfect

• To: mathgroup at smc.vnet.net
• Subject: [mg97495] ToNumberField isn't perfect
• From: Scott Morrison <scott.morrison at gmail.com>
• Date: Sat, 14 Mar 2009 05:39:28 -0500 (EST)

```I'm seeing a problem with ToNumberField sometimes claiming that a
number isn't in a number field, when it actually is.

In particular, try

d=Root[-5+17 #1^2-8 #1^4+#1^6&,6];
z=Sqrt[2-d^2];
f=Root[-5+413 #1-2156 #1^2+#1^3&,2,0];

now ToNumberField[f, z] fails, but ToNumberField[d^2 f,z]/ToNumberField
[d^2,z] (which gives the same answer) succeeds. A mathematica session
follows.

Scott Morrison

In[295]:= \[Delta]=Root[-5+17 #1^2-8 #1^4+#1^6&,6]
Out[295]= Root[-5+17 #1^2-8 #1^4+#1^6&,6]
In[296]:= z=RootReduce[Sqrt[2-\[Delta]^2]]
Out[296]= Root[-5-3 #1^2+2 #1^4+#1^6&,4]
In[297]:= f=((1-7 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2+2 Root[-5+17
#1^2-8 #1^4+#1^6&,6]^4) \[Sqrt](3-11 Root[-5+17 #1^2-8 #1^4+#1^6&,6]
^2+3 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^4) (4-11 Root[-5+17 #1^2-8
#1^4+#1^6&,6]^2+3 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^4) Root[-5+13179
#1^2+28316 #1^4+#1^6&,2])/((-2+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2) ((-1-
Root[-5+17 #1^2-8 #1^4+#1^6&,6]+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2)
(-1+Root[-5+17 #1^2-8 #1^4+#1^6&,6]+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2))
^(3/2));
In[298]:= ToNumberField[f,z]
During evaluation of In[298]:= ToNumberField::nnfel: ((1-7 Root[-5+17
Power[<<2>>]-8 Power[<<2>>]+Slot[<<1>>]^6&,6,0]^2+2 Root[-5+17 Power
[<<2>>]-8 Power[<<2>>]+Slot[<<1>>]^6&,6,0]^4) Sqrt[3-11 Root[-5+Times
[<<2>>]+Times[<<1>>]+Power[<<2>>]&,6,0]^2+3 Root[-5+Times[<<2>>]+Times
[<<2>>]+Power[<<2>>]&,6,0]^4] (4-11 Root[<<1>>&,6,0]^2+3 <<1>>^4) Root
[-5+13179 #1^2+28316 #1^4+#1^6&,2,0])/((-2+Root[-5+17 Power[<<2>>]-8
Power[<<2>>]+Slot[<<1>>]^6&,6,0]^2) ((-1-Root[-5+Times[<<2>>]+Times
[<<2>>]+Power[<<2>>]&,6,0]+Root[-5+Times[<<2>>]+Times[<<2>>]+Power
[<<2>>]&,6,0]^2) (-1+Root[<<1>>]+<<1>>^2))^(3/2)) does not belong to
the algebraic number field generated by Root[-5-3 #1^2+2 #1^4+#1^6&,
4,0]. >>
Out[298]= ToNumberField[((1-7 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2+2 Root
[-5+17 #1^2-8 #1^4+#1^6&,6]^4) Sqrt[3-11 Root[-5+17 #1^2-8 #1^4+#1^6&,
6]^2+3 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^4] (4-11 Root[-5+17 #1^2-8
#1^4+#1^6&,6]^2+3 Root[-5+17 #1^2-8 #1^4+#1^6&,6]^4) Root[-5+13179
#1^2+28316 #1^4+#1^6&,2])/((-2+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2) ((-1-
Root[-5+17 #1^2-8 #1^4+#1^6&,6]+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2)
(-1+Root[-5+17 #1^2-8 #1^4+#1^6&,6]+Root[-5+17 #1^2-8 #1^4+#1^6&,6]^2))
^(3/2)),Root[-5-3 #1^2+2 #1^4+#1^6&,4]]
In[299]:= ToNumberField[RootReduce[f],z]
During evaluation of In[299]:= ToNumberField::nnfel: Root[-5+413
#1-2156 #1^2+#1^3&,2,0] does not belong to the algebraic number field
generated by Root[-5-3 #1^2+2 #1^4+#1^6&,4,0]. >>
Out[299]= ToNumberField[Root[-5+413 #1-2156 #1^2+#1^3&,2],Root[-5-3
#1^2+2 #1^4+#1^6&,4]]
In[300]:= ToNumberField[f \[Delta]^2,z]/ToNumberField[\[Delta]^2,z]
Out[300]= AlgebraicNumber[Root[-5-3 #1^2+2 #1^4+#1^6&,4],
{554,0,668,0,183,0}]
In[301]:= N[ToNumberField[f \[Delta]^2,z]/ToNumberField[\[Delta]^2,z]-
f]
Out[301]= 1.68199*10^-14

```

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