define a field extension of the field of the rational numbers
field_extension (transcendental_elements = L, base_field = G)
field_extension (relations = J, base_field = G)
field_extension (prime_ideal = P)
list or set of names
(optional) ground field
list or set of polynomials
characterizable differential ideal
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 G⁡L 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.
K0 ≔ field_extension⁡transcendental_elements=a
K1 ≔ field_extension⁡relations=a⁢b−1,c−d,base_field=K0
R0 ≔ differential_ring⁡field_of_constants=K1,derivations=x,ranking=u
p ≔ a⁢b2⁢ux,x+c⁢u2+d3⁢ux3+1
P ≔ Rosenfeld_Groebner⁡a⁢c⁢ux−4⁢d⁢u2,R0
K2 ≔ field_extension⁡prime_ideal=P
K3 ≔ field_extension⁡transcendental_elements=e,base_field=K2
R1 ≔ differential_ring⁡field_of_constants=K3,derivations=y,ranking=v
q ≔ a⁢ux,x−8⁢u⁢ux⁢vy+b+e⁢vy,yux+x
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