Polynomials For Sums of Square Roots

For an arbitrary integer "a" the minimal polynomial with integer
coefficients and possessing the root 
                             _
                       x =  /a

can be found by squaring both sides to give

                      x^2 - a = 0

Of course, the other root of this polynomial is -sqrt(a).  If we are 
given TWO integers a,b and we wish to find a polynomial with integer 
coefficients whose roots include
                           _      _
                    x  =  /a  +  /b

we can simply square both sides to give
                                   __
                x^2  =  a + b + 2 /ab

Subtracting a+b from both sides, squaring both sides again, and
re-arranging terms gives the polynomial with integer coefficients

            [x^2 - (a+b)]^2 - 4ab  =  0

This polynomial is of degree 4, and it has the four roots give by
any combination of the two signs in
                            _      _
                  x  =  +- /a  +- /b

If we are given THREE integers a,b,c and we wish to find a polynomial
with integer coefficients whose roots include
                         _      _      _
                  x  =  /a  +  /b  +  /c

we can proceed as before, first squaring both sides and re-arranging
terms to give
                                 __    __    __
           x^2 - (a+b+c)  =  2( /ab + /ac + /bc )                (1)

Squaring both sides gives
                                              __     __     __
   [x^2 - (a+b+c)]^2  =  4[ (ab+ac+bc) + 2( a/bc + b/ac + c/ab ) ]

Letting s1 denote the first symmetric function a+b+c, and letting
s2 denote the second symmetric function ab+ac+bc, the above can be
written as
                                     __     __     __
       [x^2 - s1]^2 - 4 s2  =   8( a/bc + b/ac + c/ab )

Squaring both sides again gives
                                                     __    __    __
 [(x^2 - s1)^2 - 4 s2]^2  =  64[ abc(a+b+c) + 2abc( /ab + /ac + /bc ) ]
                                               __    __    __
                          =  64 abc [ s1 + 2( /ab + /ac + /bc ) ]

Noting that the term involving the radicals is given from equation (1)
by x^2 - s1, and letting s3 denote the third symmetric function abc,
this gives the final polynomial

        [(x^2 - s1)^2 - 4 s2]^2  -  64 s3 x^2  =  0

This is an even polynomial of degree 8 (meaning that all the coefficients 
of odd powers of x are zero), and its roots are the eight values given
by the eight possible combinations of signs in
                           _       _       _
                  x  = +- /a  +-  /b  +-  /c

Now let's suppose we are given FOUR integers a,b,c,d and we wish to
find the minimal polynomial with a root equal to
                       _      _      _      _
                x  =  /a  +  /b  +  /c  +  /d

For brevity, let sj(1) denote the jth symmetric function of the square
roots of a,b,c,d, and let sj(2) denote the jth symmetric function of 
the full values of a,b,c,d.  Our objective is to operate on the the
expression x = s1(1) until we have elimimated all the sj(1) terms
and everything is expressed in terms of sj(2) values.  We can proceed 
as before, squaring both sides of the above to give

                    x^2 = s1(2) + 2 s2(1)

Subtracting s1(2) and squaring both sides gives

      [x^2 - s1(2)]^2  =  4 s2(1)^2

                       =  4 [ s2(2) + 2{s3(1)s1(1) - s4(1)} ]

Recalling that s1(1) equals x, we can re-arrange terms to give

   [x^2 - s1(2)]^2 - 4 s2(2)  =  8 [ s3(1) x - s4(1) ]          (2)

Squaring both sides again, and noting that s4(1)^2 = s4(2), we have

   [(x^2 - s1(2))^2 - 4 s2(2)]^2  

               =  64 [ s3(1)^2 x^2 - 2s3(1)s4(1)x + s4(2) ]

Expanding s3(1)^2, we find

             s3(1)^2  =  s3(2) + 2 s4(1) s2(1)

Making this substitution and moving all the sj(2) terms over to the
left hand side, we have

 [(x^2 - s1(2))^2 - 4 s2(2)]^2 - 64[ s3(2)x^2 + s4(2) ]

                       =  128 s4(1)s2(1) x^2 - 128 s3(1)s4(1)x

We know the s2(1) equals x^2 - s1(2), and we also know from equation
(2) that

     8 s3(1) x  =  [x^2 - s1(2)]^2  -  4 s2(2)  +  8 s4(1)

Making these substitutions into the right hand side of the preceding
equation and re-arranging terms to isolate the s4(1) on the right, 
and squaring both sides again, we arrive at an expression involving
only sj(2) functions, so we will delete the parenthetical indices and
write the result as

 { [(x^2 - s1)^2 - 4 s2]^2 - 64[ s3 x^2 - s4 ] }^2

        =  s4 [ 64(x^2 - s1)x^2 - 16(x^2 - s1)^2 + 64 s2 ]^2

where
        s1 = a + b + c + d

        s2 = ab + ac + ad + bc + bd + cd

        s3 = abc + abd + acd + bcd

        s4 = abcd

To illustrate, the quantity consisting of the sum of four square roots
                   _    _    _    _
              x = /2 + /3 + /5 + /7

is a root of the polynomial

  x^16 - 136 x^14 + 6476 x^12 - 141912 x^10 + 1513334 x^8

      - 7453176 x^6  + 13950764 x^4 - 5596840 x^2 + 46225  =  0

The 16 roots of this polynomial are the values given by all 16
possible combinations of signs for the four square roots in the
expression for x.  This is the minimal polynomial with integer
coefficients having the stated value of x as a root.