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picking coefficients

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I have a general polynomial in 4 variables a,b,c,d. 

>From all terms N a^i b^j c^k d^l for i,j,k,l positive integers (all or some of i,j,k,l may be 0) with N a numerical (integer) coefficient ( N/= 0, N may be negative or positive) of the polynomial, I want to pick among these terms only those where N /== 0 mod 8.

Is it an easy way to achieve this, using Mathematica?

Example/testcase:
For Q(x) = x^4 + (2a+1) x^3 + 2b x^2 + 2c x + 2d + 1 = 0
where a, b, c and d are integers, let the polynomial
be the discriminant of the quartic Q(x) = 0.

The result of the above request should then be 5 + 4a(a+1) +4b(b+1) + 4c(c+1) + 4d(d+1), showing that the discriminant == 5 mod 8 and therefore not a square.

In fact, this shows that the Galois group of the quartic must be either Z4, D4 or S4.

Kent Holing

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picking coefficients Kent Holing <KHO@statoil.com> - 2011-04-14 09:00 +0000
  Re: picking coefficients Peter Pein <petsie@dordos.net> - 2011-04-15 07:56 +0000

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