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diffalg[field_extension] - define a field extension of the field of the rational numbers

Calling Sequence

field_extension (transcendental_elements = L, base_field = G)

field_extension (relations = J, base_field = G)

field_extension (prime_ideal = P)

Parameters

L

-

list or set of names

G

-

(optional) ground field

J

-

list or set of polynomials

P

-

characterizable differential ideal

Description

• 

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

• 

The function field_extension returns a table representing a  field extension of the field of the rational numbers. This field can be used as a field of constants for differential polynomial rings.

• 

For all the forms of field_extension, the parameter base_field = G can be omitted. In that case, it is taken as the field of the rational numbers.

• 

The first form of field_extension returns the purely transcendental field extension GL of G.

• 

The second form of field_extension returns the field of the fractions of the quotient ring G [X1 ... Xn] / (J) where the Xi are the names that appear in the polynomials of R and do not belong to G and (J) denotes the ideal generated by J in the polynomial ring G [X1 ... Xn].

  

You must ensure that the ideal (J) is prime, field_extension does not check this.

• 

The third form of field_extension returns the field of fractions of R / P where P is a characterizable differential ideal in the differential polynomial ring R.

  

You must ensure that the characterizable differential ideal P is prime. The function field_extension does not check this.

  

The embedding differential polynomial ring of P must be endowed with a jet notation.

Examples

Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.

withdiffalg:

K0:=field_extensiontranscendental_elements=a

K0:=ground_field

(1)

K1:=field_extensionrelations=ab1,cd,base_field=K0

K1:=ground_field

(2)

R0:=differential_ringfield_of_constants=K1,derivations=x,ranking=u

R0:=ODE_ring

(3)

p:=ab2ux,x+cu[]2+d3ux3+1

p:=d3ux3+ab2ux,x+cu[]2+1

(4)

reduced_formp,R0

d3ux3+ab2ux,x+du[]2+1

(5)

P:=Rosenfeld_Groebneracux4du[]2,R0

P:=characterizable

(6)

equationsP

aux4u[]2

(7)

K2:=field_extensionprime_ideal=P

K2:=ground_field

(8)

K3:=field_extensiontranscendental_elements=e,base_field=K2

K3:=ground_field

(9)

R1:=differential_ringfield_of_constants=K3,derivations=y,ranking=v

R1:=ODE_ring

(10)

q:=aux,x8u[]uxvy+b+evy,yux+x

q:=aux,x8u[]uxvy+b+evy,yux+x

(11)

reduced_formq,R1

bvy,y+evy,yux+x

(12)

See Also

diffalg(deprecated), diffalg(deprecated)/differential_algebra, diffalg(deprecated)/differential_ring, diffalg(deprecated)/reduced_form, diffalg(deprecated)/Rosenfeld_Groebner, diffalg(deprecated)[equations], DifferentialAlgebra[RosenfeldGroebner]


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