How do I write nested for-loops?

Question

Looking through the reference.wolfram, I couldn't see an example of how to write a multiple line "for" loop in Mathematica. I need to nest many for-loops in such a way I can do many things in the innermost for-loop. How does one do this in Mathematica?

Here is an example (which you can may be use to demonstrate) of the kind of thing I want to do on Mathematica. In the following, a,b,c are integers such that $1 \leq a,b,c \leq (k-1)$ and w is the m^th of the k^th roots of unity.i.e $w = exp( (2 \pi I m)/k)$ . Now for a fixed value of $k$, I am running through all values of a,b,c and $1 \leq m \leq (k-1)$. And for each set I am evaluating the roots of the polynomial, $p(x) = x^4 - 6x^2 -x(w^{a-c} + w^{c-a} + w^b + w^{-b} + w^{b-c} + w^{c-b} + w^a + w^{-a}) +(3 -w^c - w^{-c} - w^{a+b-c} - w^{-a-b+c} - w^{a-b} - w^{-a+b} )$

To check if all the roots of it are in the interval $[-2\sqrt{2},2\sqrt{2}]$. If yes, then I am printing out the value of a,b and c.

Is such a kind of nested for-loop doable in Mathematica?

---

The following code I believe runs on Sage/Python. Since this is a quartic equation, I would think that Mathematica has a way of calculating the exact roots in terms of square-roots and then doing the comparison. Also since this polynomial evaluates the eigenvalues of a Hermitian matrix, all roots should be positive.

k=6;
var('x')
for a in range(1,k):
  for b in range(1,k):
    for c in range(1,k):
      q = 1;
      for m in range (1,k):
        w = exp((2*pi*I*m )/k) 
        p(x) = 
          x^4 - 6*x^2 - x*(w^(a-c) + 
          w^(c-a) + w^b + w^(-b) + w^(b-c) + w^(c-b) + w^a + w^(-a)) + 
          (3 -w^c - w^(-c) - w^(a+b-c) - w^(-a-b+c) - w^(a-b) - w^(-a+b))
        g(x)=real_part(p(x)).simplify()
        U = (max([s.rhs() for s in g.solve(x)])).simplify_full()
        L = (min([s.rhs() for s in g.solve(x)])).simplify_full()
        if (U > 2*sqrt(2)):
          q=0
          if (L < -2*sqrt(2)): 
            q=0   
            if (q == 1):
              print "a=",a,"b=",b,"c=",c

Answer 1

It's better to stay away from loops in Mathematica:

k = 6;
l = Tuples[ConstantArray[Range[k - 1], 4]];
p[{a_, b_, c_, m_}] := Module[{w = Exp[(2*Pi*I*m)/k]}, 
      x^4 - 6*x^2 - x*(w^(a - c) + w^(c - a) + w^b + w^(-b) + w^(b - c) + w^(c - b) + 
      w^a + w^(-a)) + (3 - w^c - w^(-c) - w^(a + b - c) - w^(-a - b + c) - w^(a - b) - w^(-a + b))]
sols = Solve[p@# == 0, x] & /@ l;
compares = And @@@ (Thread[Less[Abs@x /. #, 2 Sqrt@2]] & /@ sols);
TableForm[Pick[l, compares], TableHeadings -> {None, {"a", "b", "c", "m"}}]