Dom
::AlgebraicExtension
Simple algebraic field extensions
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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)
.
If F
has Ax::canonicalRep
, then Ax::canonicalRep
.
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))

The ground field: a domain of category 

Polynomials or polynomial expressions 

Identifier 

Element of the residue class to be defined: polynomial over 

Rational function that belongs to the residue class to be defined:
expression whose numerator and denominator can be converted to polynomials
over 
"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 
"deg"  the degree of the extension, i.e., of 
"variable"  the unknown of the minimal polynomial 
"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. 