Edit detail for TypeTowerDemo revision 2 of 5

This page demonstrates some features of Axiom.

Let's begin with the construction of a polynomial ring in the indeterminate with coefficients from the ring of square matrices with entries that are polynomials where is the Galois field with 3 elements.

F:= PrimeField 3

(1)

P:=UnivariatePolynomial(x, F)

(2)

S := SquareMatrix(2, P)

(3)

R := UnivariatePolynomial(z, S)

(4)

OK, now we have the type . Let's construct an element. We start with constructing some coefficients first.

s1:S := matrix[[2*x +1 ,x^2-1],[0,x-1]]

(5)

s2 := transpose s1

(6)

And now we build the polynomial.

r: R := z^2 + s1*z + s2

(7)

Of course, since we work in characteristic 3, the following sum must be zero.

r+ 2*r

(8)

Note that this is not the integer 0, but it is still a polynomial of type .

Asking for the degree of is no problem, because is a univariate polynomial ring.

degree r

(9)

So let's see what happens if we multiply by itself.

r2 := r*r

(10)

Well, of course there is a common factor of an . Can Axiom find it?

gcd(r2, r)

   There are 4 exposed and 3 unexposed library operations named gcd
      having 2 argument(s) but none was determined to be applicable.
      Use HyperDoc Browse, or issue
                               )display op gcd
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named gcd
      with argument type(s)
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))
UnivariatePolynomial(z,SquareMatrix(2,UnivariatePolynomial(x,PrimeField 3)))

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

Ooops. What does that say?:

  Cannot find a definition or applicable library operation named gcd.

Ah, of course, the coefficient ring of is the matrix ring and this is unfortunately not a field (not even an integral domain), so we cannot simply apply Euclid's algorithm. Axiom simply stops by telling you that there is no applicaple operation.

Of course, Axiom can compute a gcd of univariate polynomials.

p1:=s1(1,1)

(11)

p2:=s1(1,2)

(12)

gcd(p1,p2)

(13)

OK, let us do that again.

q1: UP(x, INT) := 2*x+1

(14)

q2: UP(x, INT) := x^2+2

(15)

gcd(q1,q2)

(16)

Nice! Depending on where I compute these polynomials either have a common factor or are coprime.

Well, all depends on the underlying ring, of course.