Catherine Whitt

Polynomial Presentation

> a:=t^7+11*t^3-33*t+22;

> irreduc(a);

>

> irreduc(a,real);

> factor(a,real);

factor(real) gives the floating point or decimal version of the factorized polynomial, but at least it shows that the polynomial can be factored over irrational numbers, not complex numbers. If you want to factor over the complex numbers, just write complex instead of real.

> c:=t^3-7*t^2+3*t+3;

> irreduc(c);

> factor(c);

> c2:= t^2-6*t-3;

> irreduc(c2,real);

> factor(c,real);

so the polynomial c can be factored completly over the real numbers.

>

From the help page on Irreduc: The Irreduc function is a placeholder for testing the irreducibility of the multivariate polynomial. It is used in conjuction with mod and modp1.

From the help page on Factor: The Factor function is a place holder for representing the factorization of the polynomial over a unique factorization domain. It is used in conjuction with either mod or evala.

> Irreduc(t^4+7)mod 17;

> Irreduc(t^3-5)mod 11;

> Factor(t^3-5)mod 11;

Now we will use the polynomial t^4+1 and find its exact roots and its galois group.

> f:= t^4+1;

> factor(f,real);

> factor(f,complex);

> alias(alpha=RootOf(f));

> factor(f,alpha);

> convert(alpha,radical);

> solve(f,t);

> evalf(alpha);

This is -sqrt(2)/2 -sqrt(2)/2(I).

> h:= -sqrt(2)/2-(I)*sqrt(2)/2;

> expand((t-h^3)*(t+h^3)*(t-h)*(t+h));

now, we know that the galois group of a polynomial is the group of automorphisms of its splitting field. So if we take the field Q(sqrt(2),sqrt(3)), the galois group is Z (sub)2 x Z (sub) 2.

> expand((x-sqrt(2))*(x+sqrt(2))*(x-sqrt(3))*(x+sqrt(3)));

> irreduc(x^4-5*x^2+6);

> factor(x^4-5*x^2+6);

> irreduc(x^2-2);

and so x^2-3 will also not be reducible over the rationals, as expected.

> galois(x^2-2);

So it permutes its first element to its second and vice versa. This shows that x^2-2 is isomorphic to

Z (sub)2.

> galois(x^2-3);

Same thing holds for x^2-3. So individually they are the galois group of Z (sub)2, and together they make up the galois group of Z(sub)2 x Z(sub)2.