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Squares in Q[r], a question about algebraic numbers in mathematica
Posted:
Aug 8, 2012 9:36 PM


Kent,
Not having enough knowledge of algebraic fields, I don't really understand what you are doing, but in trying to follow it, I moved your Mathematica function code back into the regular global code area. Maybe the output from the code in this form will help you understand what is going on.
I do not recommend you use "Quiet" while testing, as it hides error messages which you will want to see. Also, I suggest removing the semicolons ( ; ) from your lines when trying things out in global code, so you can see the output of each line separately. (However, you must always use the semicolons inside a module).
I suggest you start every notebook with the following two lines of code:
messageHandler = If[Last[#], Interrupt[]] & ; Internal`AddHandler["Message", messageHandler];
The above will make Mathematica halt when any error message is issued.
Please paste the following code, one line at a time, into a new notebook, such that you get a separate Mathematica cell for each line. Alternatively, paste all the code into Mathematica at once, then separate each line into its own cell, using cmdD (Mac) or ctrlD (Windows).
When you test the code, run each line separately by clicking in its cell and pressing Enter. That allows you to see individual output below each input cell.
======================================== Code to be entered into Mathematica, one cell per line: ========================================
ClearAll[a7,a,p,r,pExpanded,f,q]
a7 = Root[x^4 + (5/8) x^2 + (5/8) x + (205/256), x, 1]
(*  function arguments to sq (the same function as SqrtQr)  *)
a = a7
p = (r^2 + 1)^2 /. r > a
(*  end of function arguments  *)
pExpanded = Expand[p]
f = Check[ToNumberField[Sqrt[pExpanded], a], "nonsquare"]
q = AlgebraicNumberPolynomial[f, r]
Plus[q]^2
======================================= End of code to be entered into Mathematica =======================================
For your information, following is the same code, shown as each input line followed by the output line I obtained for it. You might need to check your output from each input line against what is shown here, in case the email causes errors to appear.
========================================================= (This should not be pasted into Mathematica, as it contains extraneous text.) =========================================================
In[13]:= a7 = Root[x^4 + (5/8) x^2 + (5/8) x + (205/256), x, 1]
Out[13]= Root[205 + 160 #1 + 160 #1^2 + 256 #1^4 &, 1]
(*  function arguments to sq (which is the same function as SqrtQr)  *)
In[14]:= a = a7
Out[14]= Root[205 + 160 #1 + 160 #1^2 + 256 #1^4 &, 1]
p = (r^2 + 1)^2 /. r > a
Out[11]= (1 + Root[205 + 160 #1 + 160 #1^2 + 256 #1^4 &, 1]^2)^2
(*  end of function arguments  *)
In[15]:= pExpanded = Expand[p]
Out[15]= 1 + 2 Root[205 + 160 #1 + 160 #1^2 + 256 #1^4 &, 1]^2 + Root[205 + 160 #1 + 160 #1^2 + 256 #1^4 &, 1]^4
In[16]:= f = Check[ToNumberField[Sqrt[pExpanded], a], "nonsquare"]
Out[16]= AlgebraicNumber[ Root[205 + 40 #1 + 10 #1^2 + #1^4 &, 1], {1, 0, 1/16, 0}]
In[17]:= q = AlgebraicNumberPolynomial[f, r]
Out[17]= 1 + r^2/16
In[18]:= Plus[q]^2
Out[18]= (1 + r^2/16)^2
=========================================================
Good luck!
Ralph Dratman
On Wed, Aug 8, 2012 at 3:18 AM, Kent Holing <KHO@statoil.com> wrote: > The code below confuses me: > I thought that it should return for example (r^2+1)^2 when applied to (r^2+1)^2 whatever a is. > The code is supposed to do the following: > Given an element p (say polynomial) in Q[r] for a root r of a given polynomial in Q[x], the code should return p as q^2 for q the square root q of p as an element (polynomial) of Q[r] if q exists. > But, it does not! When applied to (r2+1)^2 it returns in fact (r^2/16+1)^2, using r/4 instead of r? > > The code: > SquareQrQ[a_,u_]:=Head@ToNumberField[Sqrt@u,a] === AlgebraicNumber//Quiet; > SqrtQr[a_,p_]:=Module[{f,q}, > f=Check[ToNumberField[Sqrt[p//Expand],a],"nonsquare"]//Quiet; > q=AlgebraicNumberPolynomial[f,r]//Quiet; > Plus[q]^2]; > sq[a_,p_]:=SqrtQr[a,p]; > > If > a=Root[x^4+5/8x^2+5/8x+205/256,x,1]; > then > sq[a,(r^2+1)^2/.r>a] returns (r^2/16+1)^2, not (r^2+1)^2. > > Any comments are highly appreciated. > Kent Holing, Norway



