// This is a comment

// This is a Boolean ring, variable x, i.e. F_2 = Z (mod 2)
// ring r = 2, x, lp;
// lp = order the monomials lexicographically
/* also a comment */


// This is how you describe galois extension fields F_{2^k}
ring r = (2, A), x, lp;
// (2, A) = 2 is the characteristic of the field, i.e. base field F_2
// A = is your alpha, root of the irreducible polynomial


// minimal polynomial of alpha
// = irreducible polynomial to construct a field
// the one below happens to be a primitive polynomial
minpoly = A^4 + A^3 + 1;

// The minpoly below is irreducible but not primitive
//minpoly = A^4 + A^3 + A^2 +A + 1;

// This is how you declare polynomials
// These are univariate polynomials with coefficients in GF(2^4)
poly f1 = (A2)*x2+(A)*x+(A3);

// In f1, A2 = A^2 = (\alpha)^2 = constant, x is the variable
// Note, Singular can take the degree as x^2, or also as x2

poly f2 = (A)*x2+(A3+A2)*x+(A);

poly f3 = (A2+A)*x2+(A3+A2+A)*x;

poly f4 = x^4 - x;


// This is how you and multiply polynomials
poly f5;

f5 = f1*f2 + (f3)^2*(f4);

// This is a C-like printf statement
printf("The computed polynomial f5 is: %s", f5);
printf("Notice that coefficients are reduced modulo A^4+A^3+1");


//////////////////////////////////////
// Conjugates

A;
A^2;
A^4;
A^8;
A^16;

// 
poly f6;
f6 = (x+A)*(x+A^2)*(x+A^4)*(x+A^8);

printf("Minimal polynomial from conjugates A, A^2, A^4, A^8 is: %s", f6);
;

printf("Compute other conjugates and construct their minimal polynomials yourself!");

// This is how you compute gcd of two elements/polynomials

poly g;
printf("GCD of f3, f4 is:");
g = gcd(f3, f4);

// Just print the GCD g
g;


// bye bye in Deutsch
quit;