       Glitch in NumberFieldFundamentalUnits

• To: mathgroup at smc.vnet.net
• Subject: [mg92373] Glitch in NumberFieldFundamentalUnits
• From: "Doug Hensley" <dahensley at suddenlink.net>
• Date: Mon, 29 Sep 2008 07:08:46 -0400 (EDT)
• References: <D0D555E836994F10AE26728DDF2AECB2@DougPC> <48DFF283.8080805@gmail.com>

```Try this:

Timing[NumberFieldFundamentalUnits[Root[x^3+4 x^2-10 x-1,1]]]

You'll get something like this:

Prepend::normal : Nonatomic expression expected at position 1 in
Prepend[System`NumberFieldDump`pvec,AlgebraicNumber[Root[-1-10 #1+4
Slot[â? 1â?¡]2+#13&,1,0],{104,-39,-10}]].

{8491.51,NumberFieldFundamentalUnits[Root[-1-10 #1+4 #12+#13&,1]]}

Now try it with another package that offers this functionality, just to get

a=bnfinit(x^3+4*x^2-10*x-1);a.fu

You'll get something like this:
(Mod(x,x^3+4x^2-10x-1),Mod(126x^2-184x-81,x^3+4x^2-10x-1)),

in far less than one second. (Meaning, that if alpha is one of the roots of
p=x^3+4x^2-10x-1, then the set {alpha,126alpha^2-184 alpha-81} comprises a
fundamental system of units for Q(alpha). )

Mathematica's NumberFieldFundamentalUnits gets fundamental unit systems for
hundreds of comparably complicated inputs, and the outputs it obtains are
not ever so much simpler in the other cases. Therefore, there's a bug. It's
the better kind of bug, because it doesn't return a wrong answer. It just
hangs and then supplies an error message.

```

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