stud_8013

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These are questions asked by stud_8013

Hello :)

So what I was trying to find Galois group for quartic polynomials in a form f=x^+a_3*x^3+a_2*x^2+a_1*x+a_0, where 0<=a_i<=2 and a_0 <>0. I get polynomials and galois groups - that's not an issue. Lets say I specifically need polynomials with group D_4 so I run script

restart;
A := {seq(0 .. 2)};
A0 := A minus {0};
m := numelems(A)^3*(numelems(A) - 1);
a := Iterator:-CartesianProduct(A0, A $ 3);
s := seq(x^4 + add(v[i]*x^(i - 1), i = 1 .. 4), v = a);
for k to m do
if irreduc(s[k]) = 'true' then print(f[k] = s[k], galois(s[k])); end if end do;

This one gives all list of polynomalias and their galois group.

for k to m do
    if irreduc(s[k]) = 'true' then if galois(s[k]) = ("4T3", {"D(4)"}, "-", 8, {"(1 3)", "(1 2 3 4)"}) then print(f[k] = s[k], galois(s[k])); end if; end if;
end do;

This one gives me a list of D_4 specifically. 
When 0<=a_i<=2 - there are not many polynomials so I cant count it by hand, but if I make a sequence larger there are too many polynomials to count them manually. So the question is how to count how many polynomials there are with a group D_4? Or lets say how do I count how many polynomials are in each group?
Thank you

Hi, I am very new in maple. I want to create a list of 54 quartic polynomials f=x^4 + a_3*x^3 + a_2*x^2 + a_1*x + a_0 with coefficients 0<=a_i<=2 and a_0 <> 0 . 

f := x ->  x^4 + add(a[i]*x^i, i = 0 .. 3);
for m to 54 do
    pol[m] := f(x);
end do;

How do I incorporate conditions 0<=a_i<=2 and a_0 <> 0 into this cycle so I can get 54 different combinations of coefficients? Thank you.

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