Squares in Q[r], a question about algebraic numbers in mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg127630] Squares in Q[r], a question about algebraic numbers in mathematica*From*: Ralph Dratman <ralph.dratman at gmail.com>*Date*: Wed, 8 Aug 2012 21:36:20 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <20120808071845.CE1DA6859@smc.vnet.net>

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 cmd-D (Mac) or ctrl-D (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], "non-square"] 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], "non-square"] 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 at 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],"non-square"]//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

**References**:**Squares in Q[r], a question about algebraic numbers in mathematica***From:*Kent Holing <KHO@statoil.com>