Should SimpleAlgebraicExtension perhaps have some more words about its
argument M. For example that M should be squarefree or irreducible?
Otherwise an element might not return 0 when plugged into its minimal
polynomial. Even worse, S claims to be a field, but it obviously has
zero divisors.
It's currently also possible to use x+5 as a defining polynomial, i.e.,
SAE also allows non-proper extensions. But that's fine for me.
Ralf
====================
(1) -> Z ==> Integer;Q==>Fraction Z
Type:
Void
(2) -> P ==> UnivariatePolynomial('x, Q)
Type:
Void
(3) -> p:P := (x^2+5)^2*(x^2+3)
6 4 2
(3) x + 13x + 55x + 75
Type:
UnivariatePolynomial(x,Fraction(Integer))
(4) -> S ==> SimpleAlgebraicExtension(Q, P, p)
Type:
Void
(5) -> s:S := generator()$S
(5) x
Type:
SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)),x^6+13*x^4+55*x^2+75)
(6) -> minimalPolynomial(s)$S
4 2
(6) x + 8x + 15
Type:
UnivariatePolynomial(x,Fraction(Integer))
(7) -> z := s^4+8*s^2+15
4 2
(7) x + 8x + 15
Type:
SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)),x^6+13*x^4+55*x^2+75)
(8) -> zero? z
(8) false
Type:
Boolean
(9) -> S has Field
(9) true
Type:
Boolean
(10) -> z*z
(10) 0
Type:
SimpleAlgebraicExtension(Fraction(Integer),UnivariatePolynomial(x,Fraction(Integer)),x^6+13*x^4+55*x^2+75)
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