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Simple algebraic field extensions
Dom::AlgebraicExtension(F, f)
Dom::AlgebraicExtension(F, f, x)
Dom::AlgebraicExtension(F, f1 = f2)
Dom::AlgebraicExtension(F, f1 = f2, x)
Dom::AlgebraicExtension(F,f)(g)
Dom::AlgebraicExtension(F, f)(rat)
For a given field F and a polynomial f ∈ F[x], Dom::AlgebraicExtension(F, f, x) creates the residue class field F[x]/<f>.
Dom::AlgebraicExtension(F, f1=f2, x) does the same for f = f_{1} - f_{2}.
Dom::AlgebraicExtension(F, f, x) creates the field F[x]/<f> of residue classes of polynomials modulo f. This field can also be written as F[x]/<f>, the field of residue classes of rational functions modulo f.
The parameter x may be omitted if f is a univariate polynomial or a polynomial expression that contains exactly one indeterminate; it is then taken to be the indeterminate occurring in f.
The field F must have normal representation.
f must not be a constant polynomial.
f must be irreducible; this is not checked.
f may be a polynomial over a coefficient ring different from F, or multivariate; however, it must be possible to convert it to a univariate polynomial over F. See Example 2.
Dom::AlgebraicExtension(F, f)(g) creates the residue class of g modulo f.
If rat has numerator and denominator p and q, respectively, then Dom::AlgebraicExtension(F,f)(rat) equals Dom::AlgebraicExtension(F,f)(p) divided by Dom::AlgebraicExtension(F,f)(q).
Cat::Field, Cat::Algebra(F), Cat::VectorSpace(F)
If F is a Cat::DifferentialRing, then Cat::DifferentialRing.
If F is a Cat::PartialDifferentialRing, then Cat::PartialDifferentialRing.
We adjoin a cubic root alpha of 2 to the rationals.
G := Dom::AlgebraicExtension(Dom::Rational, alpha^3 = 2)
The third power of a cubic root of 2 equals 2, of course.
G(alpha)^3
The trace of α is zero:
G::conjTrace(G(alpha))
You can also create random elements:
G::random()
The ground field may be an algebraic extension itself. In this way, it is possible to construct a tower of fields. In the following example, an algebraic extension is defined using a primitive element alpha, and the primitive element beta of a further extension is defined in terms of alpha. In such cases, when a minimal equation contains more than one identifier, a third argument to Dom::AlgebraicExtension must be explicitly given.
F := Dom::AlgebraicExtension(Dom::Rational, alpha^2 = 2): G := Dom::AlgebraicExtension(F, bet^2 + bet = alpha, bet)
We want to define an extension of the field of fractions of the ring of bivariate polynomials over the rationals.
P:= Dom::DistributedPolynomial([x, y], Dom::Rational): F:= Dom::Fraction(P): K:= Dom::AlgebraicExtension(F, alpha^2 = x, alpha)
Now . Of course, the square root function has the usual derivative; note that can be expressed as :
diff(K(alpha), x)
On the other hand, the derivative of with respect to y is zero, of course:
diff(K(alpha), y)
We must not use D here. This works only if we start our construction with a ring of univariate polynomials:
P:= Dom::DistributedPolynomial([x], Dom::Rational): F:= Dom::Fraction(P): K:= Dom::AlgebraicExtension(F, alpha^2 = x, alpha): D(K(alpha))
F |
The ground field: a domain of category Cat::Field |
f, f1, f2 |
Polynomials or polynomial expressions |
x |
Identifier |
g |
Element of the residue class to be defined: polynomial over F in the variable x, or any object convertible to such. |
rat |
Rational function that belongs to the residue class to be defined: expression whose numerator and denominator can be converted to polynomials over F in the variable x. The denominator must not be a multiple of f. |
"zero" | the zero element of the field extension |
"one" | the unit element of the field extension |
"groundField" | the ground field of the extension |
"minpoly" | the minimal polynomial f |
"deg" | the degree of the extension, i.e., of f |
"variable" | the unknown of the minimal polynomial f |
"characteristic" | the characteristic, which always equals the characteristic of the ground field. This entry only exists if the characteristic of the ground field is known. |
"degreeOverPrimeField" | the dimension of the field when viewed as a vector space over the prime field. This entry only exists if the ground field is a prime field, or its degree over its prime field is known. |